Cyclic Groups - Revision history

Lila: /* Subgroups of Cyclic Groups are Cyclic Groups */

2021-11-17T19:22:59Z

Subgroups of Cyclic Groups are Cyclic Groups

Lila: /* Licensing */

2021-11-17T19:21:21Z

Licensing

Lila: /* Basic Theorems Regarding Cyclic Groups */

2021-11-17T19:08:34Z

Basic Theorems Regarding Cyclic Groups

Lila at 19:06, 17 November 2021

2021-11-17T19:06:06Z

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Lila: /* Licensing */

2021-11-17T18:32:52Z

Licensing

Lila at 18:29, 17 November 2021

2021-11-17T18:29:04Z

Lila: /* Cayley graph */

2021-11-17T18:27:12Z

Cayley graph

Lila: /* Associated objects */

2021-11-17T18:26:15Z

Associated objects

Lila: /* Additional properties */

2021-11-17T18:08:06Z

Additional properties

Lila: /* Additional properties */

2021-11-17T18:07:01Z

Additional properties

@@ Line 217: / Line 217: @@
 </ul>
 <ul>
-<li>So there exists an element <span class="math-inline"><math>a^k \in H</math></span> with <span class="math-inline"><math>k &gt; 0</math></span>. Consider the set <span class="math-inline"><math>K = \{ k : a^k \in H, \; k > 0 \}</math></span>. This set is a nonempty subset of <span class="math-inline"><math>\mathbb{N}</math></span>, and so by the Well-Ordering principle, it contains a least element, call it <span class="math-inline"><math>\tilde{k}</math></span>.</li>
+<li>So there exists an element <span class="math-inline"><math>a^k \in H</math></span> with <span class="math-inline"><math>k > 0</math></span>. Consider the set <span class="math-inline"><math>K = \{ k : a^k \in H, \; k > 0 \}</math></span>. This set is a nonempty subset of <span class="math-inline"><math>\mathbb{N}</math></span>, and so by the Well-Ordering principle, it contains a least element, call it <span class="math-inline"><math>\tilde{k}</math></span>.</li>
 </ul>
 <ul>
@@ Line 238: / Line 238: @@
 <li>Thus <span class="math-inline"><math>h \in \langle a^{\tilde{k}}</math></span>, showing that <span class="math-inline"><math>H \subseteq \langle a^{\tilde{k}}</math></span>. Thus we conclude that <span class="math-inline"><math>H = \langle a^{\tilde{k}}</math></span>, i.e., <span class="math-inline"><math>H</math></span> is a cyclic group. <span class="math-inline"><math>\blacksquare</math></span></li>
 </ul>
 == Licensing ==

@@ Line 199: / Line 199: @@
 <blockquote style="background: white; border: 1px solid black; padding: 1em;"> <td><strong>Proposition 5:</strong> Let <span class="math-inline"><math>G</math></span> be a finite cyclic group of order <span class="math-inline"><math>n</math></span> with <span class="math-inline"><math>G = \langle a \rangle</math></span>. Then <span class="math-inline"><math>G</math></span> is generated by <span class="math-inline"><math>a^k</math></span> if and only if <span class="math-inline"><math>\mathrm{gcd}(n, k) = 1</math></span>.</td>
 </blockquote>
 == Licensing ==

@@ Line 165: / Line 165: @@
 <div style="text-align: center;"><math>\begin{align} \quad \varphi(g_1 \cdot g_2) = \varphi(a^{n(g_1)}a^{n(g_2)}) = \varphi(a^{n(g_1) + n(g_2)}) = n(g_1) + n(g_2) = \varphi(g_1) + \varphi(g_2) \end{align}</math></div>
 <ul>
-<li>Let <span class="math-inline"><math>g_1 = a^{n(g_1)}, g_2 = a^{n(g_2)} \in G</math></span> and suppose that <span class="math-inline"><math>\varphi(g_1) = \varphi(g_2) $}]. Then [[$ n(g_1) = n(g_2)</math></span>, so <span class="math-inline"><math>g_1 = a^{n(g_1}) = a^{n(g_2)} = g_2</math></span>, so <span class="math-inline"><math>\varphi</math></span> is injective.</li>
+<li>Let <span class="math-inline"><math>g_1 = a^{n(g_1)}, g_2 = a^{n(g_2)} \in G</math></span> and suppose that <span class="math-inline"><math>\varphi(g_1) = \varphi(g_2) </math></span>. Then <span class="math-inline"><math> n(g_1) = n(g_2)</math></span>, so <span class="math-inline"><math>g_1 = a^{n(g_1}) = a^{n(g_2)} = g_2</math></span>, so <span class="math-inline"><math>\varphi</math></span> is injective.</li>
 </ul>
 <ul>
@@ Line 185: / Line 185: @@
 <div style="text-align: center;"><math>\begin{align} \quad \varphi(a^i \cdot a^j) = \varphi (a^{i+j}) = [i] + [j]= \varphi(a^i) + \varphi(a^j) \end{align}</math></div>
 <ul>
-<li>And if <span class="math-inline"><math>a^i, a^j \in G</math></span> is such that <span class="math-inline"><math>i + j &gt; n-1</math></span> then we can write <span class="math-inline"><math>i + j = qn + r</math></span> where <span class="math-inline"><math>q \geq 1</math></span> and <span class="math-inline"><math>0 \leq r &lt; n</math></span> (by the division algorithm) so that:</li>
+<li>And if <span class="math-inline"><math>a^i, a^j \in G</math></span> is such that <span class="math-inline"><math>i + j > n-1</math></span> then we can write <span class="math-inline"><math>i + j = qn + r</math></span> where <span class="math-inline"><math>q \geq 1</math></span> and <span class="math-inline"><math>0 \leq r < n</math></span> (by the division algorithm) so that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \varphi(a^i \cdot a^j) = \varphi(a^{i+j}) = \varphi(a^{qn + r}) = \varphi((a^n)^qa^r) = \varphi(1^qa^r) = \varphi(a^r) = [r] = [qn + r] = [i +j] \end{align}</math></div>

@@ Line 147: / Line 147: @@
 Similarly, the endomorphism ring of the additive group of '''Z''' is isomorphic to the ring '''Z'''.  Its automorphism group is isomorphic to the group of units of the ring '''Z''', which is ({−1, +1}, ×) ≅ C<sub>2</sub>.
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Cyclic_group Cyclic group, Wikipedia] under a CC BY-SA license

@@ Line 31: / Line 31: @@
 |colspan=2|Two frieze groups are isomorphic to '''Z'''. With one generator, p1 has translations and p11g has glide reflections.
 |}
-On the other hand, in an '''infinite cyclic group''' ''G ='' ⟨''g''⟩'','' the powers ''g''<sup>''k''</sup> give distinct elements for all integers ''k'', so that ''G'' = { ... , ''g''<sup>&minus;2</sup>, ''g''<sup>&minus;1</sup>, ''e'', ''g'', ''g''<sup>2</sup>, ... }, and ''G'' is isomorphic to the standard group C = C<sub>∞</sub>  and to '''Z''', the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the name "cyclic" may be misleading.<ref>{{Harv|Lajoie|Mura|2000|pp=29–33}}.</ref>
+On the other hand, in an '''infinite cyclic group''' ''G ='' ⟨''g''⟩'','' the powers ''g''<sup>''k''</sup> give distinct elements for all integers ''k'', so that ''G'' = { ... , ''g''<sup>&minus;2</sup>, ''g''<sup>&minus;1</sup>, ''e'', ''g'', ''g''<sup>2</sup>, ... }, and ''G'' is isomorphic to the standard group C = C<sub>∞</sub>  and to '''Z''', the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, and the name "cyclic" may be misleading.
 To avoid this confusion, Bourbaki introduced the term '''monogenous group''' for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group".
@@ Line 37: / Line 37: @@
 ==Examples==
 ===Integer and modular addition===
-The set of integers '''Z''', with the operation of addition, forms a group.<ref name="eom"/> It is an '''infinite cyclic group''', because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to '''Z'''.
+The set of integers '''Z''', with the operation of addition, forms a group. It is an '''infinite cyclic group''', because all integers can be written by repeatedly adding or subtracting the single number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to '''Z'''.
 For every positive integer ''n'', the set of integers modulo&nbsp;''n'', again with the operation of addition, forms a finite cyclic group, denoted '''Z'''/''n'''''Z'''.
@@ Line 144: / Line 144: @@
 ===Endomorphisms===
-The endomorphism ring of the abelian group '''Z'''/''n'''''Z''' is isomorphic to '''Z'''/''n'''''Z''' itself as a ring. Under this isomorphism, the number ''r'' corresponds to the endomorphism of '''Z'''/''n'''''Z''' that maps each element to the sum of ''r'' copies of it. This is a bijection if and only if ''r'' is coprime with ''n'', so the automorphism group of '''Z'''/''n'''''Z''' is isomorphic to the unit group ('''Z'''/''n'''''Z''')<sup>×</sup>.<ref name="ks"/>
+The endomorphism ring of the abelian group '''Z'''/''n'''''Z''' is isomorphic to '''Z'''/''n'''''Z''' itself as a ring. Under this isomorphism, the number ''r'' corresponds to the endomorphism of '''Z'''/''n'''''Z''' that maps each element to the sum of ''r'' copies of it. This is a bijection if and only if ''r'' is coprime with ''n'', so the automorphism group of '''Z'''/''n'''''Z''' is isomorphic to the unit group ('''Z'''/''n'''''Z''')<sup>×</sup>.
-Similarly, the endomorphism ring of the additive group of '''Z''' is isomorphic to the ring '''Z'''.  Its automorphism group is isomorphic to the group of units of the ring '''Z''', which is {{nowrap|({−1, +1}, ×) ≅ C<sub>2</sub>}}.
+Similarly, the endomorphism ring of the additive group of '''Z''' is isomorphic to the ring '''Z'''.  Its automorphism group is isomorphic to the group of units of the ring '''Z''', which is ({−1, +1}, ×) ≅ C<sub>2</sub>.
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Cyclic_group Cyclic group, Wikipedia] under a CC BY-SA license

@@ Line 139: / Line 139: @@
 ===Cayley graph===
-{{main|Circulant graph}}
 [[File:Paley13.svg|thumb|240px|The Paley graph of order 13, a circulant graph formed as the Cayley graph of '''Z'''/13 with generator set {1,3,4}]]
 A Cayley graph is a graph defined from a pair (''G'',''S'') where ''G'' is a group and ''S'' is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs. These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the vertex-transitive graphs whose symmetry group includes a transitive cyclic group.

@@ Line 94: / Line 94: @@
 ===Representations===
-The [[Representation theory of finite groups|representation theory]] of the cyclic group is a critical base case for the representation theory of more general finite groups.  In the [[character theory|complex case]], a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent.  In the [[modular representation theory|positive characteristic case]], the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic [[Sylow subgroup]]s and more generally the representation theory of blocks of cyclic defect.
+The representation theory of the cyclic group is a critical base case for the representation theory of more general finite groups.  In the complex case, a representation of a cyclic group decomposes into a direct sum of linear characters, making the connection between character theory and representation theory transparent.  In the positive characteristic case, the indecomposable representations of the cyclic group form a model and inductive basis for the representation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocks of cyclic defect.
 ===Cycle graph===
-{{Further|Cycle graph (algebra)}}
-{{anchor|Z_sub2}}
+A '''cycle graph''' illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse generator defines a backwards path. Trivial paths (identity) can be drawn as a loop but are usually suppressed. Z<sub>2</sub> is sometimes drawn with two curved edges as a multigraph.
 A cyclic groups Z<sub>''n''</sub>, with order ''n'', corresponds to a single cycle graphed simply as an ''n''-sided polygon with the elements at the vertices.
@@ Line 141: / Line 140: @@
 ===Cayley graph===
 {{main|Circulant graph}}
-[[File:Paley13.svg|thumb|240px|The [[Paley graph]] of order 13, a circulant graph formed as the Cayley graph of '''Z'''/13 with generator set {1,3,4}]]
+[[File:Paley13.svg|thumb|240px|The Paley graph of order 13, a circulant graph formed as the Cayley graph of '''Z'''/13 with generator set {1,3,4}]]
-A [[Cayley graph]] is a graph defined from a pair (''G'',''S'') where ''G'' is a group and ''S'' is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a [[cycle graph]], and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite [[path graph]]. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called [[circulant graph]]s.<ref>{{Harv|Alspach|1997|pp=1–22}}.</ref> These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the [[vertex-transitive graph]]s whose [[Graph automorphism|symmetry group]] includes a transitive cyclic group.<ref>{{Harv|Vilfred|2004|pp=34–36}}.</ref>
+A Cayley graph is a graph defined from a pair (''G'',''S'') where ''G'' is a group and ''S'' is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs. These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point. They are exactly the vertex-transitive graphs whose symmetry group includes a transitive cyclic group.
 ===Endomorphisms===
-The endomorphism ring of the abelian group '''Z'''/''n'''''Z''' is isomorphic to '''Z'''/''n'''''Z''' itself as a ring. Under this isomorphism, the number ''r'' corresponds to the endomorphism of '''Z'''/''n'''''Z''' that maps each element to the sum of ''r'' copies of it. This is a bijection if and only if ''r'' is coprime with ''n'', so the [[automorphism group]] of '''Z'''/''n'''''Z''' is isomorphic to the unit group ('''Z'''/''n'''''Z''')<sup>×</sup>.<ref name="ks"/>
+The endomorphism ring of the abelian group '''Z'''/''n'''''Z''' is isomorphic to '''Z'''/''n'''''Z''' itself as a ring. Under this isomorphism, the number ''r'' corresponds to the endomorphism of '''Z'''/''n'''''Z''' that maps each element to the sum of ''r'' copies of it. This is a bijection if and only if ''r'' is coprime with ''n'', so the automorphism group of '''Z'''/''n'''''Z''' is isomorphic to the unit group ('''Z'''/''n'''''Z''')<sup>×</sup>.<ref name="ks"/>
 Similarly, the endomorphism ring of the additive group of '''Z''' is isomorphic to the ring '''Z'''.  Its automorphism group is isomorphic to the group of units of the ring '''Z''', which is {{nowrap|({−1, +1}, ×) ≅ C<sub>2</sub>}}.

@@ Line 89: / Line 89: @@
 :1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ...
-The definition immediately implies that cyclic groups have group presentation <math> \text{C}_\infty = \lange x | \rangle</math> and <math> \text{C}_n = \lange x | x^n \rangle </math> for finite ''n''.
+The definition immediately implies that cyclic groups have group presentation <math> \text{C}_\infty = \langle x | \rangle</math> and <math> \text{C}_n = \langle x | x^n \rangle </math> for finite ''n''.
 ==Associated objects==

@@ Line 85: / Line 85: @@
 If ''p'' is a prime number, then any group with ''p'' elements is isomorphic to the simple group '''Z'''/''p'''''Z'''.
-A number ''n''  is called a cyclic number if '''Z'''/''n'''''Z''' is the only group of order ''n'', which is true exactly when gcd(''n'',''φ''(''n'')) = 1. The cyclic numbers include all primes, but some are [[composite number|composite]] such as&nbsp;15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
+A number ''n''  is called a cyclic number if '''Z'''/''n'''''Z''' is the only group of order ''n'', which is true exactly when gcd(''n'',''φ''(''n'')) = 1. The cyclic numbers include all primes, but some are composite such as&nbsp;15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
 :1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, ...
-The definition immediately implies that cyclic groups have group presentation {{nowrap|1=C<sub>∞</sub> = {{langle}}''x'' {{!}} {{rangle}}}} and {{nowrap|1=C<sub>''n''</sub> = {{langle}}''x'' {{!}} ''x''<sup>''n''</sup>{{rangle}}}} for finite ''n''.
+The definition immediately implies that cyclic groups have group presentation <math> \text{C}_\infty = \lange x | \rangle</math> and <math> \text{C}_n = \lange x | x^n \rangle </math> for finite ''n''.
 ==Associated objects==