Abstract Algebra: 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 $0\in \mathbb {Z}$, and for all $a\in \mathbb {Z}$ we have $-a\in \mathbb {Z}$ 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 $\mathbb {Z}$ and define $*$ for all $a,b\in \mathbb {Z}$ by:

${\begin{aligned}\quad a*b=a+2b\end{aligned}}$

(Where the $+$ on the righthand side is usual addition of numbers). We will show that $(\mathbb {Z} ,*)$ is NOT a group by showing that $*$ is not associative. Let $a,b,c\in \mathbb {Z}$. Then $*$ is not associative since:

${\begin{aligned}\quad a*(b*c)=a*(b+2c)=a+2(b+2c)=a+2b+4c\end{aligned}}$

${\begin{aligned}\quad (a*b)*c=(a+2b)*c=(a+2b)+2c=a+2b+2c\end{aligned}}$

Clearly $a*(b*c)\neq (a*b)*c$ 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 *} .

Cancellation Law

We will now look at another important property of groups called the cancellation law.

Theorem 1 (The Cancellation Law for Groups): 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 a, b, c \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 = a \cdot c} 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 b \cdot a = c \cdot 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 b = c} .

• 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 a^{-1} \in G} denote 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} 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} . 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 = a \cdot c} . 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 &= a \cdot c \\ \quad (a^{-1} \cdot a) \cdot b &= (a^{-1} \cdot a) \cdot c \\ \quad e \cdot b &= e \cdot c \\ \quad b &= c \end{align}}

• Similarly, suppose now 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 b \cdot a = c \cdot 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 b \cdot a &= c \cdot a \\ \quad (b \cdot a) \cdot a^{-1} &= (c \cdot a) \cdot a^{-1} \\ \quad b \cdot (a \cdot a^{-1}) &= c \cdot (a \cdot a^{-1}) \\ \quad b \cdot e &= c \cdot e \\ \quad b &= c \quad \blacksquare \end{align}}

It is very important to note that the cancellation law holds with regards to 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} for any 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)} . We will see that the cancellation law does not necessarily hold with respect to an operation on a set when we look at algebraic structures with two defined operations.

It is also important to note 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 b = c \cdot 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 b \cdot a = a \cdot c} then we cannot necessarily deduce 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 b = c} because we would then require the additional property 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 commutative which is not one of the group axioms (but instead one of the Abelian group axioms).