Cyclic Groups

Cyclic Groups

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 for some nonnegative integer $k$.
2. If $G$ is cyclic with generator $g$ and $G$ is finite and $\left|G\right|=n$ then $G= 1, g, g2, …, gn-1 .$
3. If $G$ is cyclic then $G$ is abelian since ${g}^{i}{g}^{j}={g}^{i+j}$ for all $i,j\in ℤ$.
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, $ℤ$

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

HW: Show that $ℤ$ is an abelian group.

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

HW: Show that $ℤ$ is a cyclic group.

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

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

HW: Show that every subgroup of $ℤ$ is a normal subgroup of $ℤ$.

Example. The subgroup $5ℤ$ of the integers $ℤ$ consists of all multiples of $5$. $5ℤ= …,-10, -5,0,5, 10,…$ The subgroup $15ℤ$ is contained in the subgroup $5ℤ$. $5ℤ= …,-10, -5,0,5, 10,… ⊇ 15ℤ= …,-30, -15,0, 15,30,… .$ The sets $0+5ℤ= 5+5ℤ= …,-10, -5,0,5, 10,… =5ℤ, 1+5ℤ= -4+5ℤ= -9+5ℤ= …,-9, -4,1,6, 11,16,… , 2+5ℤ= 32+5ℤ= -23+5ℤ= …,-13, -8,-3,2, 7,12,17, 22,27,32, … , 3+5ℤ= -7+5ℤ =8+5ℤ = …,-7, -2,3, 8,13, … , 4+5ℤ= 404+5ℤ= -236+5ℤ= …,-6,-1, 4,9,14,… .$ are all cosets of the subgroup $5ℤ$ in the group $ℤ$. In fact $ℤ/5ℤ= 0+5ℤ, 1+5ℤ, 2+5ℤ, 3+5ℤ, 4+5ℤ$ is the set of cosets of $5ℤ$ in $ℤ$. As a group $ℤ/5ℤ\cong {ℤ}_{5}$.

Every homomorphism from $ℤ$ to $ℤ$ is of the form $ϕm: ℤ → ℤ n ↦ mn,$ for some positive integer m.

HW: show that $ker {\varphi }_{m}=ℤ$ 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 $ℤ$, $Aut\left(ℤ\right)=\left\{{\varphi }_{1},{\varphi }_{-1}\right\}\cong {ℤ}_{2}$

HW: Show that the inner automorphisms of $ℤ$ are $In\left(ℤ\right)=\left\{{\varphi }_{1}\right\}$

The group of integers $ℤ$ is isomorphic to the free group on one generator.

The finite cyclic groups ${C}_{n}\cong {\mu }_{n}\cong ℤ/nℤ,n\ge 1$

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 $gigj= gi+j mod n.$

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 $ℤ/nℤ$ be the group given by $ℤ/nℤ=\left\{\stackrel{‾}{0},\stackrel{‾}{1},\dots ,\stackrel{‾}{n-1}\right\}$ with operation given by $\stackrel{‾}{i}+\stackrel{‾}{j}=\stackrel{‾}{\left(i+j\right) mod n}.$ 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 $\varphi :{C}_{n}\to ℤ/nℤ$ given by $\varphi \left({g}^{i}\right)=\stackrel{‾}{i}$ 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 $Cn/ ⟨gd⟩≅ Cd/n .$

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 $H= 1, g3, g6, g9 , gH= g, g4, g7, g10 , and g2H= g2, g5, g8, g11 .$

Let ${ℂ}^{×}=ℂ\setminus \left\{0\right\}$ with the operation of multiplication of comples numbers and let $n$ be a positive integer. Every homomorphism from ${C}_{n}\to {ℂ}^{×}$ is of the from $ϕk: Cn → ℂ× g ↦ ξk , where ξ= e 2πin .$

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 $gn=1.$
2. The cyclic group ${C}_{n}$ has a presentation by generators and relations of the form $Cn= ⟨ x∣ xn=1 ⟩.$

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)