Groups - Revision history

Lila at 23:12, 16 November 2021

2021-11-16T23:12:46Z

Lila: /* Order of an element in a group */

2021-11-16T23:09:23Z

Order of an element in a group

Lila: /* Order of an element in a group */

2021-11-16T23:04:07Z

Order of an element in a group

Lila: /* Order of an element in a group */

2021-11-16T23:02:34Z

Order of an element in a group

Lila: /* Order of an element in a group */

2021-11-16T22:57:10Z

Order of an element in a group

Lila: /* Order of an element in a group */

2021-11-16T22:56:11Z

Order of an element in a group

Lila: /* Licensing */

2021-11-16T22:54:34Z

Licensing

Lila: /* Licensing */

2021-11-16T22:47:04Z

Licensing

Lila: /* Subgroups and Group Extensions */

2021-11-16T22:46:00Z

Subgroups and Group Extensions

Lila: /* Subgroups and Group Extensions */

2021-11-16T22:44:53Z

Subgroups and Group Extensions

@@ Line 177: / Line 177: @@
 <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 \: \mathrm{factors}} \cdot a = a \cdot \underbrace{(b \cdot a) \cdot (b \cdot a) \cdot ... \cdot (b \cdot a)}_{n \: \mathrm{factors}} = a \cdot (b \cdot a)^n \end{align}</math></div>
+<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>

@@ Line 154: / Line 154: @@
 <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>
 ==Licensing==

@@ Line 138: / Line 138: @@
 </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>
-<h2 id="toc1"><span>Example 1</span></h2>
+<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>
-<h2 id="toc2"><span>Example 2</span></h2>
+<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>
@@ Line 146: / Line 146: @@
 <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>
-<h2 id="toc3"><span>Example 3</span></h2>
+<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>
-<h2 id="toc4"><span>Example 4</span></h2>
+<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>
-<h2 id="toc5"><span>Example 5</span></h2>
+<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>

@@ Line 135: / Line 135: @@
 ==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>
+<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>
+<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>
 <h2 id="toc1"><span>Example 1</span></h2>
 <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>

@@ Line 135: / Line 135: @@
 ==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>
+<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>

@@ Line 132: / Line 132: @@
 <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>
 ==Licensing==

@@ Line 136: / Line 136: @@
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/groups 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

@@ Line 87: / Line 87: @@
 == Subgroups and Group Extensions==
-<p>We have just looked at the definition of a group on the <a href="/groups">Groups</a> page. Before we move on though, we will need to look at a couple of other key definitions regarding groups - namely groups that are somewhat contained within another.</p>
 <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>

@@ Line 22: / Line 22: @@
 <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>
 == Subgroups and Group Extensions==