Difference between revisions of "Groups"

Latest revision as of 17:12, 16 November 2021

Groups

Recall that an operation $\cdot$ on $S$ is said to be associative if for all $a,b,c\in S$ we have that $a\cdot (b\cdot c)=(a\cdot b)\cdot c$ and $\cdot$ is said to be commutative if for all $a,b\in S$ we have that $a\cdot b=b\cdot a$ .

An element $e\in S$ is the identity element of $S$ under $\cdot$ if for all $a\in S$ we have that $a\cdot e=a$ and $e\cdot a=a$ .

We can now begin to describe our first type of algebraic structures known as groups, which are a set $G$ equipped with a binary operation $\cdot$ that is associative, contains an identity element, and contains inverse elements under $\cdot$ for each element in $G$ .

Definition: A Group is a pair $(G,\cdot )$ where $G$ is a set and $\cdot$ is a binary operation on $G$ with the following properties:

1. For all $a,b,c\in G$ , $a\cdot (b\cdot c)=(a\cdot b)\cdot c$ (Associativity of $\cdot$ ).
2. There exists an $e\in G$ such that for all $a\in G$ , $a\cdot e=a$ and $e\cdot a=a$ (The existence of an Identity Element).
3. For all $a\in G$ there exists an $a^{-1}\in G$ such that $a\cdot a^{-1}=e$ and $a^{-1}\cdot a=e$ (The existence of inverses).

Furthermore, if $G$ is a finite set then the group $(G,\cdot )$ is said to be a Finite Group and if $G$ is an infinite set then the group $(G,\cdot )$ is said to be an Infinite Group. More generally, the Order of $(G,\cdot )$ (or **Size of $(G,\cdot )$ ) is the size of $G$ and is denoted $|G|$ .

When we use the multiplication symbol $\cdot$ to denote the operation on $G$ , we often call $G$ a “multiplicative group”. When the operation of the group is instead denoted by $+$ (instead of $\cdot$ ) then we often call $G$ an “additive group”, and we write the inverse of each $a\in G$ as $-a$ (instead of $a^{-1}$ ).

Some of the sets and binary operations we have already seen can be considered groups. For example, $(\mathbb {R} ,+)$ is a group under standard addition $+$ since the sum of any two real numbers is a real number, $+$ , is associative, an additive identity $0\in \mathbb {R}$ exists and inverse elements exist for every $a\in \mathbb {R}$ (namely $-a\in \mathbb {R}$ ).

Furthermore, $(\mathbb {Z} ,+)$ is also a group under the operation of standard addition since the sum of any two integers is an integer, addition is associative, the additivity identity is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \in \mathbb{Z}}$ , and for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in \mathbb{Z}}$ we have $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a \in \mathbb{Z}}$ as additive inverses.

We will examine many other (more interesting) groups later on, but for now, let's look at an example of a set and a binary operation that does NOT form a group.

Example 1

Consider the set of integers $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}}$ and define $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *}$ for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in \mathbb{Z}}$ by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a * b = a + 2b \end{align}}

(Where the $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +}$ on the righthand side is usual addition of numbers). We will show that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}, *)}$ is NOT a group by showing that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *}$ is not associative. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in \mathbb{Z}}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *}$ is not associative since:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a * (b * c) = a * (b + 2c) = a + 2(b + 2c) = a + 2b + 4c \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c \end{align}}

Clearly $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a * (b * c) \neq (a * b) * c}$ so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}}$ does not form a group under the operation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *}$ .

Basic Theorems Regarding Groups

A group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a set $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ with a binary operation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ such that:

1) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ is associative, i.e., for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in G}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot (b \cdot c) = (a \cdot b) \cdot c)}$ .

2) There exists an identity element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in G}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot e = a = e \cdot a}$ for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ .

3) For each $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in G}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = a^{-1} \cdot a = e}$ .

We will now look at some rather basic results regarding groups which we can derive from the group axioms above.

Proposition 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ be the identity for this group. Then:

a) The identity element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in G}$ is unique.
b) For each $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ , the corresponding inverse $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in G}$ is unique.
c) For each $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a^{-1})^{-1} = a}$ .
d) For all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in G}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^{-1} = b^{-1} \cdot a^{-1}}$ .
e) For all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in G}$ , if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b = e}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = b^{-1}}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = a^{-1}}$ .

f) If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 = a}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = e}$ .

Proof of a) Suppose that $e$ and $e'$ are both identities for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad e = e \cdot e = e \cdot e' = e' \cdot e' = e' \end{align}}

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = e'}$ so the identity for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ is unique. $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Proof of b) Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in G}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1'} \in G}$ are both inverses for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ under $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a^{-1} = a^{-1} \cdot e = a^{-1} \cdot (a \cdot a^{-1'}) = (a^{-1} \cdot a)*a^{-1} = e \cdot a^{-1'} = a^{-1'} \end{align}}

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} = a^{-1'}}$ so the inverse for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ is unique. $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Proof of c) Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a^{-1})^{-1}}$ is the inverse to $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}}$ . However, the inverse to $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ and by (b) we have shown that the inverse of each element in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is unique. Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = (a^{-1})^{-1}}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Proof of d) If we apply the operation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ between $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{-1} \cdot a^{-1}}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)}$ we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (a \cdot b) \cdot [b^{-1} \cdot a^{-1}] & = a \cdot [(b \cdot b^{-1}) \cdot a^{-1}] \\ \quad &= a \cdot [e \cdot a^{-1}] \\ \quad &= a \cdot a^{-1} \\ \quad &= e \end{align}}

Therefore the inverse of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{-1} \cdot a^{-1}}$ . We also have that the invere of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^{-1}}$ . By (b), the inverse of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)}$ is unique and so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (a \cdot b)^{-1} = b^{-1} \cdot a^{-1} \quad \blacksquare \end{align}}

Proof of e) Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b = e}$ . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a \cdot b &= e \\ \quad (a \cdot b) \cdot b^{-1} &= e \cdot b^{-1} \\ \quad a \cdot (b \cdot b^{-1}) &= b^{-1} \\ \quad a \cdot e &= b^{-1} \\ \quad a &= b^{-1} \end{align}}

Similarly:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a \cdot b &= e \\ \quad a^{-1} \cdot (a \cdot b) &= a^{-1} \cdot e \\ \quad (a^{-1} \cdot a) \cdot b &= a^{-1} \\ \quad e \cdot b &= a^{-1} \\ \quad b &= a^{-1} \quad \blacksquare \end{align}}

Proof of f) Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 = a \cdot a = a}$ . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a^2 &= a \\ \quad a \cdot a &= a \\ \quad a^{-1} \cdot (a \cdot a) &= a^{-1} \cdot a \\ \quad (a^{-1} \cdot a) \cdot a &= e \\ \quad e \cdot a &= e \\ \quad a &= e \end{align}}

Hence $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = e}$ . Alternatively we see that if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a = a}$ then the inverse of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ with respect to $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ , that is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} = e}$ . Multiplying both sides of this equation by $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ gives us that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = e}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Subgroups and Group Extensions

Definition: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group. If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ forms a group under the same operation $\cdot$ then $(S,\cdot )$ is said to be a Subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ . If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ then we write $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \leq G}$ .

Definition: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group. If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ is a group such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \subseteq H}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ is said to be a Group Extension of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ .

For example, consider the group of complex numbers under the operation of standard addition, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{C}, +)}$ . We know that the set of real numbers is a subset of the set of complex numbers, that is, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} \subset \mathbb{C}}$ and so the group of real numbers under the operation of standard addition, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}, +)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{C}, +)}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{C}, +)}$ is a group extension of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}, +)}$ .

We will now look at a nice theorem which tells us that to determine if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ , that then we only need to check two of the four group axioms for verification

Theorem 1: If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a group with the identity $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in G}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ if and only if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is closed under $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in S}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = e}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \cdot a = e}$ .

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group with the identity $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in G}$ of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ .

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}$ Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ . Then by definition, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a group itself and satisfies all of the group axioms - namely that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is closed under the operation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and that for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in S}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = e}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \cdot a = e}$ .

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow}$ Now suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is closed under $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and that for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in S}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = e}$ and $a^{-1}\cdot a=e$ . These are precisely two of the group axioms we have looked at, and to show that $(S,\cdot )$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ we only need to show that the other two axioms hold.

First suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in S}$ and that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot (b \cdot c) \neq (a \cdot b) \cdot c}$ , that is, suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ is not associative on $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ . Since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ we must have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot (b \cdot c) \neq (a \cdot b) \cdot c}$ for this particular $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in G}$ which contradicts the associativity of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ on the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ . Hence $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ must actually be associative on $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ .

Now since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is closed under $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in S}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = e}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \cdot a = e}$ we must have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in S}$ and furthermore, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot e = a}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \cdot a = a}$ .

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a group, and in particular since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Theorem 2: If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a group with the identity $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in G}$ of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subseteq G}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ if and only if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \neq \emptyset}$ and for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in H}$ we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b^{-1} \in H}$ .

Proof: $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}$ If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot) } is a subgroup of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ then this direction is trivial.

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow}$ Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \neq \emptyset}$ and for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in S}$ we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b^{-1} \in S}$ . Since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \neq \emptyset}$ there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot a^{-1} = e \in S}$ .

Now if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in S}$ , then since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in S}$ we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \cdot a^{-1} = a^{-1} \in S}$ . So if $a\in S$ then $a^{-1}\in H$ .

Lastly, if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in S}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b^{-1} \in S}$ . Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot (b^{-1})^{-1} = a \cdot b \in S}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}$ is closed under the operation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot}$ . Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S, \cdot)}$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Order of an element in a group

Definition: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, *)}$ be a group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ . The Order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ denoted by $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a|}$ or $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{ord}(a)}$ is the smallest positive integer $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^m = \underbrace{a \cdot a \cdot ... \cdot a}_{m : \text{many factors}} = e}$ (where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ is the identity element of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ ). If no such $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}$ exists then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ is said to have order $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}$ .

If the operation is multiplicative in nature then we usually define the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ as above. If the operation is instead additive in nature then we define the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ as the smallest positive integer $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ma = \underbrace{(a + a + ... + a)}_{m \; \text{many}} = e}$ or $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}$ if no such positive integer $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}$ exists.

Example 1

If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is any group with identity $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ then the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}$ .

Example 2

Consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}_5, +)}$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +}$ is defined for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in \mathbb{Z}_5 = \{ 0, 1, 2, 3, 4 \}}$ to be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad x + y = (x + y) \mod 5 \end{align}}

The order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = 0}$ is trivially $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}$ . The order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5}$ since:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 1^5 = (1 + 1 + 1 + 1 + 1) = 0 \mod 5 \end{align}}

The order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}$ is also $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5}$ . In fact, the orders of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4}$ are also $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5}$ .

Example 3

Consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}_6, +)}$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +}$ is defined for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in \mathbb{Z}_6 = \{ 0, 1, 2, 3, 4, 5 \}}$ to be:

{\begin{aligned}\quad x+y=(x+y)\mod 6\end{aligned}}

You should verify that the order of $0$ is $1$ , the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6}$ , the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}$ , the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}$ , the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}$ , and the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6}$

Example 4

Consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R}, +)}$ . Then every nonzero $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}}$ has order infinity since the equation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle mx = 0}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \geq 1}$ , has no solution in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} \setminus \{ 0 \}}$ .

Example 5

Consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R} \setminus \{ 0 \}, \cdot)}$ . The order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}$ since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1)^2 = 1}$ .

Basic Theorems Regarding the Order of Elements in a Group

Proposition 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group. Then:

a) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{order}(a) = 1}$ if and only if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = e}$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}$ is the identity element of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ .
b) If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a^{-1}) = \mathrm{ord}(a)}$ .

c) If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in G}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a \cdot b) = \mathrm{ord}(b \cdot a)}$ . _

Proof of a) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}$ Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a)}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^1 = e}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = e}$ .

$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow}$ The smallest positive integer such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^n = e}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(e) = 1}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Proof of b) Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G}$ . Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ has finite order, say $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a) = n}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}$ is the smallest positive integer such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^n = e}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-n} = e}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a^{-1}) \leq n}$ . If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a^{-1}) = m < n}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-m} = e}$ implies that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^m = e}$ , and since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m < n = \mathrm{ord}(a)}$ we have arrived at a contradiction. Thus $\mathrm {ord} (a^{-1})=\mathrm {ord} (a)$ .

Now suppose that $a$ has infinite order. If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}}$ has finite order, say $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a^{-1}) = m < \infty}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-m} = e}$ implies that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^m = e}$ - contradicting $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}$ having infinite order. Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}}$ must also have infinite order.

Proof of c) Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in G}$ . First, suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a \cdot b) = n < \infty}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}$ is the smallest positive integer such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^n = e}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^n \cdot a = a}$ . Now observe that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (a \cdot b)^n \cdot a = \underbrace{(a \cdot b) \cdot (a \cdot b) \cdot ... \cdot (a \cdot b)}_{n \; \text{factors}} \cdot a = a \cdot \underbrace{(b \cdot a) \cdot (b \cdot a) \cdot ... \cdot (b \cdot a)}_{n \; \text{factors}} = a \cdot (b \cdot a)^n \end{align}}

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^n \cdot a = a \cdot (b \cdot a)^n}$ . But $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^n = e}$ , so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = a \cdot (b \cdot a)^n}$ . Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b \cdot a)^n = e}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(b \cdot a) \leq n = \mathrm{ord}(a \cdot b)}$ .

By symmetry, we see that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a \cdot b) \leq \mathrm{ord}(b \cdot a)}$ . Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a \cdot b) = \mathrm{ord}(b \cdot a)}$ .

Now suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(a \cdot b)}$ is infinite. If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(b \cdot a) = n < \infty}$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b \cdot a)^n = e}$ . By the same argument above, we see that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \cdot b)^n = e}$ - contradicting $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b}$ having infinite order. Thus $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}(b \cdot a)}$ is infinite. $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

Licensing

Content obtained and/or adapted from:

Groups, mathonline.wikidot.com under a CC BY-SA license
Basic Theorems Regarding Groups, mathonline.wikidot.com under a CC BY-SA license
Subgroups and Group Extensions, mathonline.wikidot.com under a CC BY-SA license

@@ Line 1: / Line 1: @@
+==Groups==
 <p>Recall that an operation <span class="math-inline"><math>\cdot</math></span> on <span class="math-inline"><math>S</math></span> is said to be associative if for all <span class="math-inline"><math>a, b, c \in S</math></span> we have that <span class="math-inline"><math>a \cdot (b \cdot c) = (a \cdot b) \cdot c</math></span> and <span class="math-inline"><math>\cdot</math></span> is said to be commutative if for all <span class="math-inline"><math>a, b \in S</math></span> we have that <span class="math-inline"><math>a \cdot b = b \cdot a</math></span>.</p>
 <p>An element <span class="math-inline"><math>e \in S</math></span> is the identity element of <span class="math-inline"><math>S</math></span> under <span class="math-inline"><math>\cdot</math></span> if for all <span class="math-inline"><math>a \in S</math></span> we have that <span class="math-inline"><math>a \cdot e = a</math></span> and <span class="math-inline"><math>e \cdot a = a</math></span>.</p>
@@ Line 21: / Line 23: @@
 <p>Clearly <span class="math-inline"><math>a * (b * c) \neq (a * b) * c</math></span> so <span class="math-inline"><math>\mathbb{Z}</math></span> does not form a group under the operation <span class="math-inline"><math>*</math></span>.</p>
+==Basic Theorems Regarding Groups==
+<p>A group <span class="math-inline"><math>(G, \cdot)</math></span> is a set <span class="math-inline"><math>G</math></span> with a binary operation <span class="math-inline"><math>\cdot</math></span> such that:</p>
+<ul>
+<li><strong>1)</strong> <span class="math-inline"><math>\cdot</math></span> is associative, i.e., for all <span class="math-inline"><math>a, b, c \in G</math></span>, <span class="math-inline"><math>a \cdot (b \cdot c) = (a \cdot b) \cdot c)</math></span>.</li>
+</ul>
+<ul>
+<li><strong>2)</strong> There exists an identity element <span class="math-inline"><math>e \in G</math></span> such that <span class="math-inline"><math>a \cdot e = a = e \cdot a</math></span> for all <span class="math-inline"><math>a \in G</math></span>.</li>
+</ul>
+<ul>
+<li><strong>3)</strong> For each <span class="math-inline"><math>a \in G</math></span> there exists an <span class="math-inline"><math>a^{-1} \in G</math></span> such that <span class="math-inline"><math>a \cdot a^{-1} = a^{-1} \cdot a = e</math></span>.</li>
+</ul>
+<p>We will now look at some rather basic results regarding groups which we can derive from the group axioms above.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Proposition 1:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group and let <span class="math-inline"><math>e</math></span> be the identity for this group. Then:<br />
+<strong>a)</strong> The identity element <span class="math-inline"><math>e \in G</math></span> is unique.<br />
+<strong>b)</strong> For each <span class="math-inline"><math>a \in G</math></span>, the corresponding inverse <span class="math-inline"><math>a^{-1} \in G</math></span> is unique.<br />
+<strong>c)</strong> For each <span class="math-inline"><math>a \in G</math></span>, <span class="math-inline"><math>(a^{-1})^{-1} = a</math></span>.<br />
+<strong>d)</strong> For all <span class="math-inline"><math>a, b \in G</math></span>, <span class="math-inline"><math>(a \cdot b)^{-1} = b^{-1} \cdot a^{-1}</math></span>.<br />
+<strong>e)</strong> For all <span class="math-inline"><math>a, b \in G</math></span>, if <span class="math-inline"><math>a \cdot b = e</math></span> then <span class="math-inline"><math>a = b^{-1}</math></span> and <span class="math-inline"><math>b = a^{-1}</math></span>.<br />
+<strong>f)</strong> If <span class="math-inline"><math>a^2 = a</math></span> then <span class="math-inline"><math>a = e</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof of a)</strong> Suppose that <span class="math-inline"><math>e</math></span> and <span class="math-inline"><math>e'</math></span> are both identities for <span class="math-inline"><math>\cdot</math></span>. Then:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad e = e \cdot e = e \cdot e' = e' \cdot e' = e' \end{align}</math></div>
+<ul>
+<li>Therefore <span class="math-inline"><math>e = e'</math></span> so the identity for <span class="math-inline"><math>\cdot</math></span> is unique. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<ul>
+<li><strong>Proof of b)</strong> Suppose that <span class="math-inline"><math>a^{-1} \in G</math></span> and <span class="math-inline"><math>a^{-1'} \in G</math></span> are both inverses for <span class="math-inline"><math>a \in G</math></span> under <span class="math-inline"><math>\cdot</math></span>. Then:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad a^{-1} = a^{-1} \cdot e = a^{-1} \cdot (a \cdot a^{-1'}) = (a^{-1} \cdot a)*a^{-1} = e \cdot a^{-1'} = a^{-1'} \end{align}</math></div>
+<ul>
+<li>Therefore <span class="math-inline"><math>a^{-1} = a^{-1'}</math></span> so the inverse for <span class="math-inline"><math>a</math></span> is unique. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<ul>
+<li><strong>Proof of c)</strong> Let <span class="math-inline"><math>a \in G</math></span>. Then <span class="math-inline"><math>(a^{-1})^{-1}</math></span> is the inverse to <span class="math-inline"><math>a^{-1}</math></span>. However, the inverse to <span class="math-inline"><math>a^{-1}</math></span> is <span class="math-inline"><math>a</math></span> and by (b) we have shown that the inverse of each element in <span class="math-inline"><math>G</math></span> is unique. Therefore <span class="math-inline"><math>a = (a^{-1})^{-1}</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<ul>
+<li><strong>Proof of d)</strong> If we apply the operation <span class="math-inline"><math>\cdot</math></span> between <span class="math-inline"><math>b^{-1} \cdot a^{-1}</math></span> and <span class="math-inline"><math>(a \cdot b)</math></span> we get:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad (a \cdot b) \cdot [b^{-1} \cdot a^{-1}] & = a \cdot [(b \cdot b^{-1}) \cdot a^{-1}] \\ \quad &= a \cdot [e \cdot a^{-1}] \\ \quad &= a \cdot a^{-1} \\ \quad &= e \end{align}</math></div>
+<ul>
+<li>Therefore the inverse of <span class="math-inline"><math>(a \cdot b)</math></span> is <span class="math-inline"><math>b^{-1} \cdot a^{-1}</math></span>. We also have that the invere of <span class="math-inline"><math>(a \cdot b)</math></span> is <span class="math-inline"><math>(a \cdot b)^{-1}</math></span>. By (b), the inverse of <span class="math-inline"><math>(a \cdot b)</math></span> is unique and so:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad (a \cdot b)^{-1} = b^{-1} \cdot a^{-1} \quad \blacksquare \end{align}</math></div>
+<ul>
+<li><strong>Proof of e)</strong> Suppose that <span class="math-inline"><math>a \cdot b = e</math></span>. Then:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad a \cdot b &= e \\ \quad (a \cdot b) \cdot b^{-1} &= e \cdot b^{-1} \\ \quad a \cdot (b \cdot b^{-1}) &= b^{-1} \\ \quad a \cdot e &= b^{-1} \\ \quad a &= b^{-1} \end{align}</math></div>
+<ul>
+<li>Similarly:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad a \cdot b &= e \\ \quad a^{-1} \cdot (a \cdot b) &= a^{-1} \cdot e \\ \quad (a^{-1} \cdot a) \cdot b &= a^{-1} \\ \quad e \cdot b &= a^{-1} \\ \quad b &= a^{-1} \quad \blacksquare \end{align}</math></div>
+<ul>
+<li><strong>Proof of f)</strong> Suppose that <span class="math-inline"><math>a^2 = a \cdot a = a</math></span>. Then:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad a^2 &= a \\ \quad a \cdot a &= a \\ \quad a^{-1} \cdot (a \cdot a) &= a^{-1} \cdot a \\ \quad (a^{-1} \cdot a) \cdot a &= e \\ \quad e \cdot a &= e \\ \quad a &= e \end{align}</math></div>
+<ul>
+<li>Hence <span class="math-inline"><math>a = e</math></span>. Alternatively we see that if <span class="math-inline"><math>a \cdot a = a</math></span> then the inverse of <span class="math-inline"><math>a</math></span> with respect to <span class="math-inline"><math>\cdot</math></span> is <span class="math-inline"><math>e</math></span>, that is <span class="math-inline"><math>a^{-1} = e</math></span>. Multiplying both sides of this equation by <span class="math-inline"><math>a</math></span> gives us that <span class="math-inline"><math>a = e</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+== Subgroups and Group Extensions==
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Definition:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group. If <span class="math-inline"><math>S \subseteq G</math></span> and <span class="math-inline"><math>S</math></span> forms a group under the same operation <span class="math-inline"><math>\cdot</math></span> then <span class="math-inline"><math>(S, \cdot)</math></span> is said to be a <strong>Subgroup</strong> of <span class="math-inline"><math>(G, \cdot)</math></span>. If <span class="math-inline"><math>S</math></span> is a subgroup of <span class="math-inline"><math>G</math></span> then we write <span class="math-inline"><math>S \leq G</math></span>.</td>
+</blockquote>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Definition:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group. If <span class="math-inline"><math>(H, \cdot)</math></span> is a group such that <span class="math-inline"><math>G \subseteq H</math></span> then <span class="math-inline"><math>(H, \cdot)</math></span> is said to be a <strong>Group Extension</strong> of <span class="math-inline"><math>(G, \cdot)</math></span>.</td>
+</blockquote>
+<p>For example, consider the group of complex numbers under the operation of standard addition, <span class="math-inline"><math>(\mathbb{C}, +)</math></span>. We know that the set of real numbers is a subset of the set of complex numbers, that is, <span class="math-inline"><math>\mathbb{R} \subset \mathbb{C}</math></span> and so the group of real numbers under the operation of standard addition, <span class="math-inline"><math>(\mathbb{R}, +)</math></span> is a subgroup of <span class="math-inline"><math>(\mathbb{C}, +)</math></span> and <span class="math-inline"><math>(\mathbb{C}, +)</math></span> is a group extension of <span class="math-inline"><math>(\mathbb{R}, +)</math></span>.</p>
+<p>We will now look at a nice theorem which tells us that to determine if <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span> where <span class="math-inline"><math>S \subseteq G</math></span>, that then we only need to check two of the four group axioms for verification</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1:</strong> If <span class="math-inline"><math>(G, \cdot)</math></span> is a group with the identity <span class="math-inline"><math>e \in G</math></span> and <span class="math-inline"><math>S \subseteq G</math></span> then <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span> if and only if <span class="math-inline"><math>S</math></span> is closed under <span class="math-inline"><math>\cdot</math></span> and for all <span class="math-inline"><math>a \in S</math></span> there exists an <span class="math-inline"><math>a^{-1} \in S</math></span> such that <span class="math-inline"><math>a \cdot a^{-1} = e</math></span> and <span class="math-inline"><math>a^{-1} \cdot a = e</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group with the identity <span class="math-inline"><math>e \in G</math></span> of <span class="math-inline"><math>\cdot</math></span> and let <span class="math-inline"><math>S \subseteq G</math></span>.</li>
+</ul>
+<ul>
+<li><span class="math-inline"><math>\Rightarrow</math></span> Suppose that <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span>. Then by definition, <span class="math-inline"><math>(S, \cdot)</math></span> is a group itself and satisfies all of the group axioms - namely that <span class="math-inline"><math>S</math></span> is closed under the operation <span class="math-inline"><math>\cdot</math></span> and that for all <span class="math-inline"><math>a \in S</math></span> there exists an <span class="math-inline"><math>a^{-1} \in S</math></span> such that <span class="math-inline"><math>a \cdot a^{-1} = e</math></span> and <span class="math-inline"><math>a^{-1} \cdot a = e</math></span>.</li>
+</ul>
+<ul>
+<li><span class="math-inline"><math>\Leftarrow</math></span> Now suppose that <span class="math-inline"><math>S</math></span> is closed under <span class="math-inline"><math>\cdot</math></span> and that for all <span class="math-inline"><math>a \in S</math></span> there exists an <span class="math-inline"><math>a^{-1} \in S</math></span> such that <span class="math-inline"><math>a \cdot a^{-1} = e</math></span> and <span class="math-inline"><math>a^{-1} \cdot a = e</math></span>. These are precisely two of the group axioms we have looked at, and to show that <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span> we only need to show that the other two axioms hold.</li>
+</ul>
+<ul>
+<li>First suppose that <span class="math-inline"><math>a, b, c \in S</math></span> and that <span class="math-inline"><math>a \cdot (b \cdot c) \neq (a \cdot b) \cdot c</math></span>, that is, suppose that <span class="math-inline"><math>\cdot</math></span> is not associative on <span class="math-inline"><math>S</math></span>. Since <span class="math-inline"><math>S \subseteq G</math></span> we must have that <span class="math-inline"><math>a \cdot (b \cdot c) \neq (a \cdot b) \cdot c</math></span> for this particular <span class="math-inline"><math>a, b, c \in G</math></span> which contradicts the associativity of <span class="math-inline"><math>\cdot</math></span> on the group <span class="math-inline"><math>G</math></span>. Hence <span class="math-inline"><math>\cdot</math></span> must actually be associative on <span class="math-inline"><math>G</math></span>.</li>
+</ul>
+<ul>
+<li>Now since <span class="math-inline"><math>S</math></span> is closed under <span class="math-inline"><math>\cdot</math></span> and for <span class="math-inline"><math>a \in S</math></span> there exists an <span class="math-inline"><math>a^{-1} \in S</math></span> such that <span class="math-inline"><math>a \cdot a^{-1} = e</math></span> and <span class="math-inline"><math>a^{-1} \cdot a = e</math></span> we must have that <span class="math-inline"><math>e \in S</math></span> and furthermore, <span class="math-inline"><math>a \cdot e = a</math></span> and <span class="math-inline"><math>e \cdot a = a</math></span>.</li>
+</ul>
+<ul>
+<li>Therefore <span class="math-inline"><math>(S, \cdot)</math></span> is a group, and in particular since <span class="math-inline"><math>S \subseteq G</math></span> we have that <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 2:</strong> If <span class="math-inline"><math>(G, \cdot)</math></span> is a group with the identity <span class="math-inline"><math>e \in G</math></span> of <span class="math-inline"><math>\cdot</math></span> and <span class="math-inline"><math>S \subseteq G</math></span> then <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span> if and only if <span class="math-inline"><math>H \neq \emptyset</math></span> and for all <span class="math-inline"><math>a, b \in H</math></span> we have that <span class="math-inline"><math>a \cdot b^{-1} \in H</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> <span class="math-inline"><math>\Rightarrow</math></span> If <span class="math-inline"><math>(S, \cdot) </math> is a subgroup of <math> (G, \cdot)</math></span> then this direction is trivial.</li>
+</ul>
+<ul>
+<li><span class="math-inline"><math>\Leftarrow</math></span> Suppose that <span class="math-inline"><math>S \neq \emptyset</math></span> and for all <span class="math-inline"><math>a, b \in S</math></span> we have that <span class="math-inline"><math>a \cdot b^{-1} \in S</math></span>. Since <span class="math-inline"><math>S \neq \emptyset</math></span> there exists an <span class="math-inline"><math>a \in S</math></span>. So <span class="math-inline"><math>a \cdot a^{-1} = e \in S</math></span>.</li>
+</ul>
+<ul>
+<li>Now if <span class="math-inline"><math>a \in S</math></span>, then since <span class="math-inline"><math>e \in S</math></span> we have that <span class="math-inline"><math>e \cdot a^{-1} = a^{-1} \in S</math></span>. So if <span class="math-inline"><math>a \in S</math></span> then <span class="math-inline"><math>a^{-1} \in H</math></span>.</li>
+</ul>
+<ul>
+<li>Lastly, if <span class="math-inline"><math>a, b \in S</math></span> then <span class="math-inline"><math>a, b^{-1} \in S</math></span>. Thus <span class="math-inline"><math>a \cdot (b^{-1})^{-1} = a \cdot b \in S</math></span>. So <span class="math-inline"><math>S</math></span> is closed under the operation <span class="math-inline"><math>\cdot</math></span>. Thus <span class="math-inline"><math>(S, \cdot)</math></span> is a subgroup of <span class="math-inline"><math>(G, \cdot)</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+==Order of an element in a group==
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Definition:</strong> Let <span class="math-inline"><math>(G, *)</math></span> be a group and let <span class="math-inline"><math>a \in G</math></span>. The <strong>Order</strong> of <span class="math-inline"><math>a</math></span> denoted by <span class="math-inline"><math>|a|</math></span> or <span class="math-inline"><math>\text{ord}(a)</math></span> is the smallest positive integer <span class="math-inline"><math>m</math></span> such that <span class="math-inline"><math>a^m = \underbrace{a \cdot a \cdot ... \cdot a}_{m : \text{many factors}} = e</math></span> (where <span class="math-inline"><math>e</math></span> is the identity element of <span class="math-inline"><math>G</math></span>). If no such <span class="math-inline"><math>m</math></span> exists then <span class="math-inline"><math>a</math></span> is said to have order <span class="math-inline"><math>\infty</math></span>.</td>
+</blockquote>
+<p><em>If the operation is multiplicative in nature then we usually define the order of <span class="math-inline"><math>a \in G</math></span> as above. If the operation is instead additive in nature then we define the order of <span class="math-inline"><math>a \in G</math></span> as the smallest positive integer <span class="math-inline"><math>m</math></span> such that <span class="math-inline"><math>ma = \underbrace{(a + a + ... + a)}_{m \; \text{many}} = e</math></span> or <span class="math-inline"><math>\infty</math></span> if no such positive integer <span class="math-inline"><math>m</math></span> exists.</em></p>
+<h3 id="toc1"><span>Example 1</span></h3>
+<p>If <span class="math-inline"><math>G</math></span> is any group with identity <span class="math-inline"><math>e</math></span> then the order of <span class="math-inline"><math>e</math></span> is <span class="math-inline"><math>1</math></span>.</p>
+<h3 id="toc2"><span>Example 2</span></h3>
+<p>Consider the group <span class="math-inline"><math>(\mathbb{Z}_5, +)</math></span> where <span class="math-inline"><math>+</math></span> is defined for all <span class="math-inline"><math>x, y \in \mathbb{Z}_5 = \{ 0, 1, 2, 3, 4 \}</math></span> to be:</p>
+<div style="text-align: center;"><math>\begin{align} \quad x + y = (x + y) \mod 5 \end{align}</math></div>
+<p>The order of <span class="math-inline"><math>e = 0</math></span> is trivially <span class="math-inline"><math>1</math></span>. The order of <span class="math-inline"><math>1</math></span> is <span class="math-inline"><math>5</math></span> since:</p>
+<div style="text-align: center;"><math>\begin{align} \quad 1^5 = (1 + 1 + 1 + 1 + 1) = 0 \mod 5 \end{align}</math></div>
+<p>The order of <span class="math-inline"><math>2</math></span> is also <span class="math-inline"><math>5</math></span>. In fact, the orders of <span class="math-inline"><math>3</math></span> and <span class="math-inline"><math>4</math></span> are also <span class="math-inline"><math>5</math></span>.</p>
+<h3 id="toc3"><span>Example 3</span></h3>
+<p>Consider the group <span class="math-inline"><math>(\mathbb{Z}_6, +)</math></span> where <span class="math-inline"><math>+</math></span> is defined for all <span class="math-inline"><math>x, y \in \mathbb{Z}_6 = \{ 0, 1, 2, 3, 4, 5 \}</math></span> to be:</p>
+<div style="text-align: center;"><math>\begin{align} \quad x + y = (x + y) \mod 6 \end{align}</math></div>
+<p>You should verify that the order of <span class="math-inline"><math>0</math></span> is <span class="math-inline"><math>1</math></span>, the order of <span class="math-inline"><math>1</math></span> is <span class="math-inline"><math>6</math></span>, the order of <span class="math-inline"><math>2</math></span> is <span class="math-inline"><math>3</math></span>, the order of <span class="math-inline"><math>3</math></span> is <span class="math-inline"><math>2</math></span>, the order of <span class="math-inline"><math>4</math></span> is <span class="math-inline"><math>3</math></span>, and the order of <span class="math-inline"><math>5</math></span> is <span class="math-inline"><math>6</math></span></p>
+<h3 id="toc4"><span>Example 4</span></h3>
+<p>Consider the group <span class="math-inline"><math>(\mathbb{R}, +)</math></span>. Then every nonzero <span class="math-inline"><math>x \in \mathbb{R}</math></span> has order infinity since the equation <span class="math-inline"><math>mx = 0</math></span>, <span class="math-inline"><math>m \geq 1</math></span>, has no solution in <span class="math-inline"><math>\mathbb{R} \setminus \{ 0 \}</math></span>.</p>
+<h3 id="toc5"><span>Example 5</span></h3>
+<p>Consider the group <span class="math-inline"><math>(\mathbb{R} \setminus \{ 0 \}, \cdot)</math></span>. The order of <span class="math-inline"><math>-1</math></span> is <span class="math-inline"><math>2</math></span> since <span class="math-inline"><math>(-1)^2 = 1</math></span>.</p>
+==Basic Theorems Regarding the Order of Elements in a Group==
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Proposition 1:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group. Then:<br />
+<strong>a)</strong> <span class="math-inline"><math>\mathrm{order}(a) = 1</math></span> if and only if <span class="math-inline"><math>a = e</math></span> where <span class="math-inline"><math>e</math></span> is the identity element of <span class="math-inline"><math>G</math></span>.<br />
+<strong>b)</strong> If <span class="math-inline"><math>a \in G</math></span> then <span class="math-inline"><math>\mathrm{ord}(a^{-1}) = \mathrm{ord}(a)</math></span>.<br />
+<strong>c)</strong> If <span class="math-inline"><math>a, b \in G</math></span> then <span class="math-inline"><math>\mathrm{ord}(a \cdot b) = \mathrm{ord}(b \cdot a)</math></span>. _</td>
+</blockquote>
+<ul>
+<li><strong>Proof of a)</strong> <span class="math-inline"><math>\Rightarrow</math></span> Suppose that <span class="math-inline"><math>\mathrm{ord}(a)</math></span>. Then <span class="math-inline"><math>a^1 = e</math></span>. So <span class="math-inline"><math>a = e</math></span>.</li>
+</ul>
+<ul>
+<li><span class="math-inline"><math>\Leftarrow</math></span> The smallest positive integer such that <span class="math-inline"><math>e^n = e</math></span> is <span class="math-inline"><math>n = 1</math></span>. So <span class="math-inline"><math>\mathrm{ord}(e) = 1</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<ul>
+<li><strong>Proof of b)</strong> Let <span class="math-inline"><math>a \in G</math></span>. Suppose that <span class="math-inline"><math>a</math></span> has finite order, say <span class="math-inline"><math>\mathrm{ord}(a) = n</math></span>. Then <span class="math-inline"><math>n</math></span> is the smallest positive integer such that <span class="math-inline"><math>a^n = e</math></span>. So <span class="math-inline"><math>a^{-n} = e</math></span>. So <span class="math-inline"><math>\mathrm{ord}(a^{-1}) \leq n</math></span>. If <span class="math-inline"><math>\mathrm{ord}(a^{-1}) = m < n</math></span> then <span class="math-inline"><math>a^{-m} = e</math></span> implies that <span class="math-inline"><math>a^m = e</math></span>, and since <span class="math-inline"><math>m < n = \mathrm{ord}(a)</math></span> we have arrived at a contradiction. Thus <span class="math-inline"><math>\mathrm{ord}(a^{-1}) = \mathrm{ord}(a)</math></span>.</li>
+</ul>
+<ul>
+<li>Now suppose that <span class="math-inline"><math>a</math></span> has infinite order. If <span class="math-inline"><math>a^{-1}</math></span> has finite order, say <span class="math-inline"><math>\mathrm{ord}(a^{-1}) = m < \infty</math></span> then <span class="math-inline"><math>a^{-m} = e</math></span> implies that <span class="math-inline"><math>a^m = e</math></span> - contradicting <span class="math-inline"><math>a</math></span> having infinite order. Thus <span class="math-inline"><math>a^{-1}</math></span> must also have infinite order.</li>
+</ul>
+<ul>
+<li><strong>Proof of c)</strong> Let <span class="math-inline"><math>a, b \in G</math></span>. First, suppose that <span class="math-inline"><math>\mathrm{ord}(a \cdot b) = n < \infty</math></span>. Then <span class="math-inline"><math>n</math></span> is the smallest positive integer such that <span class="math-inline"><math>(a \cdot b)^n = e</math></span>. So <span class="math-inline"><math>(a \cdot b)^n \cdot a = a</math></span>. Now observe that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad (a \cdot b)^n \cdot a = \underbrace{(a \cdot b) \cdot (a \cdot b) \cdot ... \cdot (a \cdot b)}_{n \; \text{factors}} \cdot a = a \cdot \underbrace{(b \cdot a) \cdot (b \cdot a) \cdot ... \cdot (b \cdot a)}_{n \; \text{factors}} = a \cdot (b \cdot a)^n \end{align}</math></div>
+<ul>
+<li>Therefore <span class="math-inline"><math>(a \cdot b)^n \cdot a = a \cdot (b \cdot a)^n</math></span>. But <span class="math-inline"><math>(a \cdot b)^n = e</math></span>, so <span class="math-inline"><math>a = a \cdot (b \cdot a)^n</math></span>. Thus <span class="math-inline"><math>(b \cdot a)^n = e</math></span>. So <span class="math-inline"><math>\mathrm{ord}(b \cdot a) \leq n = \mathrm{ord}(a \cdot b)</math></span>.</li>
+</ul>
+<ul>
+<li>By symmetry, we see that <span class="math-inline"><math>\mathrm{ord}(a \cdot b) \leq \mathrm{ord}(b \cdot a)</math></span>. Thus <span class="math-inline"><math>\mathrm{ord}(a \cdot b) = \mathrm{ord}(b \cdot a)</math></span>.</li>
+</ul>
+<ul>
+<li>Now suppose that <span class="math-inline"><math>\mathrm{ord}(a \cdot b)</math></span> is infinite. If <span class="math-inline"><math>\mathrm{ord}(b \cdot a) = n < \infty</math></span> then <span class="math-inline"><math>(b \cdot a)^n = e</math></span>. By the same argument above, we see that <span class="math-inline"><math>(a \cdot b)^n = e</math></span> - contradicting <span class="math-inline"><math>a \cdot b</math></span> having infinite order. Thus <span class="math-inline"><math>\mathrm{ord}(b \cdot a)</math></span> is infinite. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
 ==Licensing==
 Content obtained and/or adapted from:
-* [] under a CC BY-SA license
+* [http://mathonline.wikidot.com/groups Groups, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/basic-theorems-regarding-groups Basic Theorems Regarding Groups, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/subgroups-and-group-extensions Subgroups and Group Extensions, mathonline.wikidot.com] under a CC BY-SA license

Difference between revisions of "Groups"

Latest revision as of 17:12, 16 November 2021

Contents

Groups

Example 1

Basic Theorems Regarding Groups

Subgroups and Group Extensions

Order of an element in a group

Example 1

Example 2

Example 3

Example 4

Example 5

Basic Theorems Regarding the Order of Elements in a Group

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