Cyclic Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 6 December 2010

Definition. A cyclic group is a group $G$ that contains an element $g\in G$ such that the group generated by $g$ is $G$, $⟨g⟩=G$.

The following facts follow from the definition

1. If $G$ is cyclic with generator $g$ then all elements of $g$ are of the form $$g^k=\underset{\underset{k\text{factors}}{⏟}}{\phantom{(}g\cdot g\cdots g}\text{or}{g}^{-k}=\underset{\underset{k\text{factors}}{⏟}}{g^{-1}\cdot g^{-1}\cdots g^{-1}}$$ for some nonnegative integer $k$.
2. If $G$ is cyclic with generator $g$ and $G$ is finite and $\left|G\right|=n$ then $$G=\left\{1,g,g^2,\dots ,g^{n-1}\right\}.$$
3. If $G$ is cyclic then $G$ is abelian since $g^i{g}^j=g^{i+j}$ for all $i,j\in \mathbb{Z}$.
4. If $G$ is cyclic then all subgroups of $G$ are normal since $G$ is abelian.

HW: Let $G$ be a group of order $p$ where $p$ is prime. Show that $G$ is cyclic.

The integers, $\mathbb{Z}$

Definition. The group of integers $\mathbb{Z}$ is the set $\mathbb{Z}=\left\{\dots ,-2,-1,0,1,2,\dots \right\}$ with the operation of addition.

HW: Show that $\mathbb{Z}$ is an abelian group.

HW: Show that both the element $1\in \mathbb{Z}$ and the element $-1\in \mathbb{Z}$ generate $\mathbb{Z}$.

HW: Show that $\mathbb{Z}$ is a cyclic group.

HW: Show that every element of $\mathbb{Z}$ is in a conjugacy class by itself.

1. Let $H$ be a subset of the integers $\mathbb{Z}$. Then $H$ is a subgroup of $\mathbb{Z}$ if and only if $H=m\mathbb{Z}$ for some nonnegative integer $m$.
2. Let $m$ and $n$ be positive integers. Then $m\mathbb{Z}\subseteq n\mathbb{Z}$ if and only if $n$ divides $m$.
3. Let $n$ be a positive integer. Then the quotient group $\mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}/n\mathbb{Z}.$

HW: Show that every subgroup of $\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}$.

Example. The subgroup $5\mathbb{Z}$ of the integers $\mathbb{Z}$ consists of all multiples of $5$. $$5\mathbb{Z}=\left\{\dots ,-10,-5,0,5,10,\dots \right\}$$ The subgroup $15\mathbb{Z}$ is contained in the subgroup $5\mathbb{Z}$. $$\begin{array}{ccc}5\mathbb{Z}=\left\{\dots ,-10,-5,0,5,10,\dots \right\}& \supseteq & 15\mathbb{Z}=\left\{\dots ,-30,-15,0,15,30,\dots \right\}.\end{array}$$ The sets $$\begin{array}{l}0+5\mathbb{Z}=5+5\mathbb{Z}=\left\{\dots ,-10,-5,0,5,10,\dots \right\}=5\mathbb{Z},\\ 1+5\mathbb{Z}=-4+5\mathbb{Z}=-9+5\mathbb{Z}=\left\{\dots ,-9,-4,1,6,11,16,\dots \right\},\\ 2+5\mathbb{Z}=32+5\mathbb{Z}=-23+5\mathbb{Z}=\left\{\dots ,-13,-8,-3,2,7,12,17,22,27,32,\dots \right\},\\ 3+5\mathbb{Z}=-7+5\mathbb{Z}=8+5\mathbb{Z}=\left\{\dots ,-7,-2,3,8,13,\dots \right\},\\ 4+5\mathbb{Z}=404+5\mathbb{Z}=-236+5\mathbb{Z}=\left\{\dots ,-6,-1,4,9,14,\dots \right\}.\end{array}$$ are all cosets of the subgroup $5\mathbb{Z}$ in the group $\mathbb{Z}$. In fact $$\mathbb{Z}/5\mathbb{Z}=\left\{0+5\mathbb{Z},1+5\mathbb{Z},2+5\mathbb{Z},3+5\mathbb{Z},4+5\mathbb{Z}\right\}$$ is the set of cosets of $5\mathbb{Z}$ in $\mathbb{Z}$. As a group $\mathbb{Z}/5\mathbb{Z}\cong {\mathbb{Z}}_5$.

Every homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$ is of the form $$\begin{array}{cccc}{\varphi }_m:& \mathbb{Z}& \to & \mathbb{Z}\\ & n& \mapsto & mn,\end{array}$$ for some positive integer m.

HW: show that $ker\hspace{0.17em}{\varphi }_m=\mathbb{Z}$ if $m=0$.

HW: Show that ${\varphi }_m$ is injective if $m\ne 0$.

HW: Show that ${\varphi }_m$ is bijective if and only $m=1$ or $m=-1$.

HW: Show that ${\varphi }_1$ is the identity mapping.

HW: Show that the automorphism group of $\mathbb{Z}$, $Aut\left(\mathbb{Z}\right)=\left\{{\varphi }_1,{\varphi }_{-1}\right\}\cong {\mathbb{Z}}_2$

HW: Show that the inner automorphisms of $\mathbb{Z}$ are $In\left(\mathbb{Z}\right)=\left\{{\varphi }_1\right\}$

The group of integers $\mathbb{Z}$ is isomorphic to the free group on one generator.

The finite cyclic groups $C_n\cong {\mu }_n\cong \mathbb{Z}/n\mathbb{Z},n\ge 1$

Definition. The cyclic group of order $n$, $C_n$ is the set $\left\{1,g,g^2,\dots ,g^{n-1}\right\}$ with the operation given by $$g^i{g}^j=g^{i+j\hspace{0.17em}mod\hspace{0.17em}n}.$$

There are other representations of the group $C_n$ which are useful.

1. Let ${\mu }_n$ be the group given by ${\mu }_n=\left\{1,\xi ,{\xi }^2,\dots ,{\xi }^{n-1}\right\}$ where $\xi =e^{\frac{2\pi i}{n}}\in ℂ,$ with the operation of multiplication of complex numbers. In the complex plane the elements of ${\mu }_n$ all lie on the circle $S^1=\left\{z\in ℂ\mid \left|z\right|=1\right\}.$

2. Let $\mathbb{Z}/n\mathbb{Z}$ be the group given by $\mathbb{Z}/n\mathbb{Z}=\left\{\bar{0},\bar{1},\dots ,\overline{n-1}\right\}$ with operation given by $\bar{i}+\bar{j}=\overline{\left(i+j\right)\hspace{0.17em}mod\hspace{0.17em}n}.$ This operation is called addition modulo $n$.

HW: Show that the group homomorphism $\phi :C_n\to {\mu }_n$ given by $\phi \left(g^i\right)={\xi }^i$ is an isomorphism.

HW: Show that the group homomorphism $\varphi :C_n\to \mathbb{Z}/n\mathbb{Z}$ given by $\varphi \left(g^i\right)=\bar{i}$ is an isomorphism.

1. The subgropus of the cyclic group $C_n$ are the subgroups generated by the elements $g^m$, $⟨g^m⟩$, $0\le m\le n-1$.
2. Let $0\le m\le n-1$ and let $d=gcd\left(m,n\right)$. Then $⟨g^m⟩=⟨g^d⟩$ and $\left|⟨g^d⟩\right|=n/d.$
3. Let $0\le m,k\le n-1.$ Then $⟨g^n⟩\subseteq ⟨g^k⟩$ if and only if $gcd\left(k,n\right)$ divides $gcd\left(m,n\right).$
4. Let $0\le d\le n$ and suppose that $d$ divides $n$. Then $$C_n/⟨g^d⟩\cong C_{d/n}.$$

Example: The subgroup lattice of the group $C_{12}$ is given by

The set of cosets of $C_{12}/⟨g^3⟩=\left\{H,gH,g^{2}H\right\}$ where $$H=\left\{1,g^3,g^6,g^9\right\},gH=\left\{g,g^4,g^7,g^{10}\right\},\text{and}{g}^{2}H=\left\{g^2,g^5,g^8,g^{11}\right\}.$$

Let ${ℂ}^{\times }=ℂ\setminus \left\{0\right\}$ with the operation of multiplication of comples numbers and let $n$ be a positive integer. Every homomorphism from $C_n\to {ℂ}^{\times }$ is of the from $$\begin{array}{cccc}{\varphi }_k:& C_n& \to & {ℂ}^{\times }\\ & g& \mapsto & {\xi }^k\end{array},\text{where}\xi =e^{\frac{2\pi i}{n}}.$$

1. The cyclic group $C_n=\left\{1,g,g^2,\dots ,g^{n-1}\right\}$ of order $n$ is generated by the element $g$ and $g$ satisfies $$g^n=1.$$
2. The cyclic group $C_n$ has a presentation by generators and relations of the form $$C_n=⟨x\mid x^n=1⟩.$$

Let $S$ be a circular necklace with $n$ equally spaced beads $b_0,b_1,\dots ,b_{n-1},$ numbered counterclockwise around $S$.

1. There is an action of the cyclic group $C_n$ on the necklace $S$ such that $g$ acts by rotating $S$ counterclockwise by an angle of $2\pi /n$.
2. This action has one orbit, $C_n{b}_0=\left\{b_0,b_1,\dots ,b_{n-1}\right\}$ and the stabiliser of each bead is the group $⟨1⟩$.

References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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