Abstract Algebra: Groups - Revision history

Lila: /* Cancellation Law */

2021-11-19T20:28:14Z

Cancellation Law

Lila at 20:05, 19 November 2021

2021-11-19T20:05:01Z

Lila at 19:57, 19 November 2021

2021-11-19T19:57:47Z

Lila: Created page with "
Recall that an operation $\cdot$ on $S$ is said to be associative if for all
2021-11-19T19:40:50Z

Created page with "Recall that an operation <math>\cdot</math> on <math>S</math> is said to be associative if for all Recall that an operation <math>\cdot</math> on <math>S</math> is said to be associative if for all <math>a, b, c \in S</math> we have that <math>a \cdot (b \cdot c) = (a \cdot b) \cdot c</math> and <math>\cdot</math> is said to be commutative if for all <math>a, b \in S</math> we have that <math>a \cdot b = b \cdot a</math>.
An element <math>e \in S</math> is the identity element of <math>S</math> under <math>\cdot</math> if for all <math>a \in S</math> we have that <math>a \cdot e = a</math> and <math>e \cdot a = a</math>.
We can now begin to describe our first type of algebraic structures known as groups, which are a set <math>G</math> equipped with a binary operation <math>\cdot</math> that is associative, contains an identity element, and contains inverse elements under <math>\cdot</math> for each element in <math>G</math>.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: A Group is a pair <math>(G, \cdot)</math> where <math>G</math> is a set and <math>\cdot</math> is a binary operation on <math>G</math> with the following properties: 
1. For all <math>a, b, c \in G</math>, <math>a \cdot (b \cdot c) = (a \cdot b) \cdot c</math> (Associativity of <math>\cdot</math>). 
2. There exists an <math>e \in G</math> such that for all <math>a \in G</math>, <math>a \cdot e = a</math> and <math>e \cdot a = a</math> (The existence of an Identity Element). 
3. For all <math>a \in G</math> there exists an <math>a^{-1} \in G</math> such that <math>a \cdot a^{-1} = e</math> and <math>a^{-1} \cdot a = e</math> (The existence of inverses). 
Furthermore, if <math>G</math> is a finite set then the group <math>(G, \cdot)</math> is said to be a Finite Group and if <math>G</math> is an infinite set then the group <math>(G, \cdot)</math> is said to be an Infinite Group. More generally, the Order of <math>(G, \cdot)</math> (or **Size of <math>(G, \cdot)</math>) is the size of <math>G</math> and is denoted <math>| G |</math>.</td>
</blockquote>
When we use the multiplication symbol <math>\cdot</math> to denote the operation on <math>G</math>, we often call <math>G</math> a “multiplicative group”. When the operation of the group is instead denoted by <math>+</math> (instead of <math>\cdot</math>) then we often call <math>G</math> an “additive group”, and we write the inverse of each <math>a \in G</math> as <math>-a</math> (instead of <math>a^{-1}</math>).
Some of the sets and binary operations we have already seen can be considered groups. For example, <math>(\mathbb{R}, +)</math> is a group under standard addition <math>+</math> since the sum of any two real numbers is a real number, <math>+</math>, is associative, an additive identity <math>0 \in \mathbb{R}</math> exists and inverse elements exist for every <math>a \in \mathbb{R}</math> (namely <math>-a \in \mathbb{R}</math>).
Furthermore, <math>(\mathbb{Z}, +)</math> 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 <math>0 \in \mathbb{Z}</math>, and for all <math>a \in \mathbb{Z}</math> we have <math>-a \in \mathbb{Z}</math> 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 <math>\mathbb{Z}</math> and define <math></math> for all <math>a, b \in \mathbb{Z}</math> by:
<div style="text-align: center;"><math>\begin{align} \quad a b = a + 2b \end{align}</math></div>
(Where the <math>+</math> on the righthand side is usual addition of numbers). We will show that <math>(\mathbb{Z}, )</math> is NOT a group by showing that <math></math> is not associative. Let <math>a, b, c \in \mathbb{Z}</math>. Then <math></math> is not associative since:
<div style="text-align: center;"><math>\begin{align} \quad a (b * c) = a * (b + 2c) = a + 2(b + 2c) = a + 2b + 4c \end{align}</math></div>
<div style="text-align: center;"><math>\begin{align} \quad (a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c \end{align}</math></div>
Clearly <math>a * (b * c) \neq (a * b) * c</math> so <math>\mathbb{Z}</math> does not form a group under the operation <math>*</math>.

← Older revision		Revision as of 20:28, 19 November 2021
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	<p>It is very important to note that the cancellation law holds with regards to the operation <span class="math-inline"><math>\cdot</math></span> for any group <span class="math-inline"><math>(G, \cdot)</math></span>. 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.</p>		<p>It is very important to note that the cancellation law holds with regards to the operation <span class="math-inline"><math>\cdot</math></span> for any group <span class="math-inline"><math>(G, \cdot)</math></span>. 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.</p>
	<p>It is also important to note that if <span class="math-inline"><math>a \cdot b = c \cdot a</math></span> or <span class="math-inline"><math>b \cdot a = a \cdot c</math></span> then we cannot necessarily deduce that <span class="math-inline"><math>b = c</math></span> because we would then require the additional property that <span class="math-inline"><math>\cdot</math></span> is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).</p>		<p>It is also important to note that if <span class="math-inline"><math>a \cdot b = c \cdot a</math></span> or <span class="math-inline"><math>b \cdot a = a \cdot c</math></span> then we cannot necessarily deduce that <span class="math-inline"><math>b = c</math></span> because we would then require the additional property that <span class="math-inline"><math>\cdot</math></span> is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).</p>
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [http://mathonline.wikidot.com/abstract-algebra Abstract Algebra, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 21: / Line 21: @@
 <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>
-===Cancellation Law===
+==Basic Theorems Regarding Groups==
 <p>We will now look at another important property of groups called the cancellation law.</p>
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">

← Older revision		Revision as of 19:57, 19 November 2021
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	<div style="text-align: center;"><math>\begin{align} \quad (a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c \end{align}</math></div>		<div style="text-align: center;"><math>\begin{align} \quad (a * b) * c = (a + 2b) * c = (a + 2b) + 2c = a + 2b + 2c \end{align}</math></div>
	<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>		<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>
		+
		+	===Cancellation Law===
		+	<p>We will now look at another important property of groups called the cancellation law.</p>
		+	<blockquote style="background: white; border: 1px solid black; padding: 1em;">
		+	<td><strong>Theorem 1 (The Cancellation Law for Groups):</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group and let <span class="math-inline"><math>a, b, c \in G</math></span>. If <span class="math-inline"><math>a \cdot b = a \cdot c</math></span> or <span class="math-inline"><math>b \cdot a = c \cdot a</math></span> then <span class="math-inline"><math>b = c</math></span>.</td>
		+	</blockquote>
		+	<ul>
		+	<li><strong>Proof:</strong> Let <span class="math-inline"><math>a^{-1} \in G</math></span> denote the inverse of <span class="math-inline"><math>a</math></span> under <span class="math-inline"><math>\cdot</math></span>. Suppose that <span class="math-inline"><math>a \cdot b = a \cdot c</math></span>. Then:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\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}</math></div>
		+	<ul>
		+	<li>Similarly, suppose now that <span class="math-inline"><math>b \cdot a = c \cdot a</math></span>. Then:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\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}</math></div>
		+	<p>It is very important to note that the cancellation law holds with regards to the operation <span class="math-inline"><math>\cdot</math></span> for any group <span class="math-inline"><math>(G, \cdot)</math></span>. 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.</p>
		+	<p>It is also important to note that if <span class="math-inline"><math>a \cdot b = c \cdot a</math></span> or <span class="math-inline"><math>b \cdot a = a \cdot c</math></span> then we cannot necessarily deduce that <span class="math-inline"><math>b = c</math></span> because we would then require the additional property that <span class="math-inline"><math>\cdot</math></span> is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).</p>