## Group Theory and Linear Algebra

Last updated: 24 September 2014

## Lecture 21: Cyclic groups and products

The general linear group with entries in $ℂ$ is $GLn(ℂ) = {n×n matrices g with entries in ℂ such that g is invertible} = { g∈Mn(ℂ) | g-1∈Mn(ℂ) } = { g∈Mn(ℂ) | det(g)≠0 } .$ So $GL2(ℂ)= { (abcd) | a,b,c,d∈ℂ and ad-bc≠0 }$ with product matrix multiplication. So $GL1(ℂ) = {g∈M1(ℂ) | det(g)≠0} = {c∈ℂ | c≠0} = ℂ-{0} = ℂ×.$

A cyclic group is a group generated by one element.

$ℤ3ℤ={0,1,2} is generated byS={1}. {,,} is generated byS={}. {,,,} is generated byS={}. {,,,,} is generated byS={}. {1,g,g2,g3,g4} withg5=1 is generated byg.$ In this last example $g3g4=g7= g5g2=1· g2=g2.$

$\left\{1,\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\right\}$ (a subset of $ℂ\text{)}$ is a group under multiplication. If $\zeta =\frac{-1+\sqrt{3}i}{2}$ then ${\zeta }^{2}=\frac{-1-\sqrt{3}i}{2}$ and ${\zeta }^{3}=1$ so that ${1,-1+3i2,-1-3i2} ={1,ζ,ζ2}with ζ3=1$ and this group is generated by $S=\left\{\zeta \right\}=\left\{\frac{-1+\sqrt{3}i}{2}\right\}\text{.}$

The group of ${n}^{\text{th}}$ roots of unity is $μn={z∈ℂ | zn=1}.$ The group of ${3}^{\text{rd}}$ roots of unity is $\left\{1,\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\right\}={\mu }_{3}\text{.}$ Then ${\mu }_{n}=\left\{z\in ℂ | {z}^{n}=1\right\}$ is a subgroup of ${GL}_{1}\left(ℂ\right)={ℂ}^{×}$ and ${\mu }_{n}$ is generated by $\eta ={e}^{2\pi i/n}$ where $e2πi/n= cos(2πn)+ isin(2πn).$ Note: $e2πi/3= cos(2π3)+ isin(2π3)= -12+i32= -1+3i2.$

Let $G$ and $H$ be groups. The product of $G$ and $H$ is the set $G×H={(g,h) | g∈G and h∈H}$ with $(g1,h1) (g2,h2)= (g1g2,h1h2).$

$ℤ}{2ℤ}=\left\{0,1\right\}$ and $ℤ2ℤ×ℤ2ℤ ={(0,0),(0,1),(1,0),(1,1)}$ with $(0,0) (0,1) (1,0) (1,1) (0,0) (0,0) (0,1) (1,0) (1,1) (0,1) (0,1) (0,0) (1,1) (1,0) (1,0) (1,0) (1,1) (0,0) (0,1) (1,1) (1,1) (1,0) (0,1) (0,0) Order of ℤ2ℤ×ℤ2ℤ is 4 Order of (0,0) is 1 Order of (1,0) is 2 Order of (0,1) is 2 Order of (1,1) is 2$

$\left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\}$ is a subgroup of ${S}_{4}\text{.}$ $Order of {,,,} is 4 Order of is 1 Order of is 2 Order of is 2 Order of is 2$ and the function $ℤ2ℤ×ℤ2ℤ ⟶ {,,,} (0,0) ⟼ (1,0) ⟼ (0,1) ⟼ (1,1) ⟼$ is an isomorphism.

Subgroups of $ℤ}{2ℤ}×ℤ}{2ℤ}\text{:}$ $ℤ}{2ℤ}×ℤ}{2ℤ} \left\{\left(0,0\right),\left(1,0\right)\right\} \left\{\left(0,0\right),\left(0,1\right)\right\} \left\{\left(0,0\right),\left(1,1\right)\right\} \left\{\left(0,0\right)\right\}$ Subgroups of $\left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\}:$ $\left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array}\right\}$

## Notes and References

These are a typed copy of Lecture 21 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on September 9, 2011.