Difference between revisions of "Groups"

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==Groups==
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<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>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>
 
<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>
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<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>
  
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== Subgroups and Group Extensions==
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<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>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<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>
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</blockquote>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<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>
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</blockquote>
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<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>
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<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>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<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>
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</blockquote>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<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>
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</blockquote>
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<ul>
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<li><strong>Proof:</strong> <span class="math-inline"><math>\Rightarrow</math></span> If <span class="math-inline"><math>(S, \cdot) $] is a subgroup of [[$ (G, \cdot)</math></span> then this direction is trivial.</li>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
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<ul>
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<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>
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</ul>
  
 
==Licensing==
 
==Licensing==
 
Content obtained and/or adapted from:
 
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/groups Groups, mathonline.wikidot.com] under a CC BY-SA license

Revision as of 16:31, 16 November 2021

Groups

Recall that an 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} 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} is said to be associative if for all we have that and is said to be commutative if for all we have that .

An element is the identity element of under if 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} 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 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} .

We can now begin to describe our first type of algebraic structures known as groups, which are 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} equipped 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} that is associative, contains an identity element, and contains inverse elements 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} for 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} .

Definition: A Group is a pair Failed to parse (MathML with SVG or PNG fallback (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 G} is a set 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 \cdot} is a binary operation 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} with the following properties:

1. 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} (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} ).
2. 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 e \in G} such 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 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 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} (The existence of an Identity Element).
3. 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} 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} = 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} (The existence of inverses).

Furthermore, 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 a finite set then 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, \cdot)} is said to be a Finite Group and 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 an infinite set then 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, \cdot)} is said to be an Infinite Group. More generally, 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 (G, \cdot)} (or **Size 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)} ) is the size 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} and is denoted Failed to parse (MathML with SVG or PNG fallback (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 |} .

When we use the multiplication symbol Failed to parse (MathML with SVG or PNG fallback (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} to denote the operation 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} , we often call Failed to parse (MathML with SVG or PNG fallback (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} a “multiplicative group”. When the operation of the group is instead 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 +} (instead 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} ) then we often call Failed to parse (MathML with SVG or PNG fallback (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} an “additive group”, and we write the inverse of 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} as Failed to parse (MathML with SVG or PNG fallback (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} (instead 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^{-1}} ).

Some of the sets and binary operations we have already seen can be considered groups. For example, Failed to parse (MathML with SVG or PNG fallback (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 group under standard addition since the sum of any two real numbers is a real number, , is associative, an additive identity exists and inverse elements exist for every Failed to parse (MathML with SVG or PNG fallback (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{R}} (namely Failed to parse (MathML with SVG or PNG fallback (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{R}} ).

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 (\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 *} .

Subgroups and Group Extensions

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.

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 Failed to parse (MathML with SVG or PNG fallback (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 (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 if and only if 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 Failed to parse (MathML with SVG or PNG fallback (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} . These are precisely two of the group axioms we have looked at, and to 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 (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 is a group, and in particular since 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) $] is a subgroup of [[$ (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 Failed to parse (MathML with SVG or PNG fallback (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 Failed to parse (MathML with SVG or PNG fallback (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 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}

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