Group Theory and Linear Algebra

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

Last updated: 24 September 2014

Lecture 21: Cyclic groups and products

The general linear group with entries in $ℂ$ is $$\begin{array}{ccc}{\displaystyle {GL}_n\left(ℂ\right)}& =& {\displaystyle \{n\times n\hspace{0.17em}\text{matrices}\hspace{0.17em}g\hspace{0.17em}\text{with entries in}\hspace{0.17em}ℂ\hspace{0.17em}\text{such that}\hspace{0.17em}g\hspace{0.17em}\text{is invertible}\}}\\ & =& {\displaystyle \{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{g}^{-1}\in M_n\left(ℂ\right)\}}\\ & =& {\displaystyle \{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)\ne 0\}\text{.}}\end{array}$$ So $${GL}_2\left(ℂ\right)=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}a,b,c,d\in ℂ\hspace{0.17em}\text{and}\hspace{0.17em}ad-bc\ne 0\}$$ with product matrix multiplication. So $$\begin{array}{ccc}{\displaystyle {GL}_1\left(ℂ\right)}& =& {\displaystyle \{g\in M_1\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)\ne 0\}}\\ & =& {\displaystyle \{c\in ℂ\hspace{0.17em}|\hspace{0.17em}c\ne 0\}}\\ & =& {\displaystyle ℂ-\left\{0\right\}}\\ & =& {\displaystyle {ℂ}^{\times }\text{.}}\end{array}$$

A cyclic group is a group generated by one element.

$$\begin{array}{c}\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$3\mathbb{Z}$}\right.=\{0,1,2\}\text{is generated by}S=\left\{1\right\}\text{.}\\ \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\text{is generated by}S=\left\{\begin{array}{c}\end{array}\right\}\text{.}\\ \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\text{is generated by}S=\left\{\begin{array}{c}\end{array}\right\}\text{.}\\ \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\text{is generated by}S=\left\{\begin{array}{c}\end{array}\right\}\text{.}\\ \{1,g,g^2,g^3,g^4\}\text{with}{g}^5=1\text{is generated by}g\text{.}\end{array}$$ In this last example $$g^3{g}^4=g^7=g^5{g}^2=1·g^2=g^2\text{.}$$

$\{1,\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\}$ (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 $$\{1,\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\}=\{1,\zeta ,{\zeta }^2\}\text{with}{\zeta }^3=1$$ 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 $${\mu }_n=\{z\in ℂ\hspace{0.17em}|\hspace{0.17em}{z}^n=1\}\text{.}$$ The group of $3^{\text{rd}}$ roots of unity is $\{1,\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\}={\mu }_3\text{.}$ Then ${\mu }_n=\{z\in ℂ\hspace{0.17em}|\hspace{0.17em}{z}^n=1\}$ is a subgroup of ${GL}_1\left(ℂ\right)={ℂ}^{\times }$ and ${\mu }_n$ is generated by $\eta =e^{2\pi i/n}$ where $$e^{2\pi i/n}=\text{cos}\left(\frac{2\pi }{n}\right)+i\text{sin}\left(\frac{2\pi }{n}\right)\text{.}$$ Note: $$e^{2\pi i/3}=\text{cos}\left(\frac{2\pi }{3}\right)+i\text{sin}\left(\frac{2\pi }{3}\right)=-\frac{1}{2}+i\frac{\sqrt{3}}{2}=\frac{-1+\sqrt{3}i}{2}\text{.}$$

Let $G$ and $H$ be groups. The product of $G$ and $H$ is the set $$G\times H=\left\{(g,h)\hspace{0.17em}\right|\hspace{0.17em}g\in G\hspace{0.17em}\text{and}\hspace{0.17em}h\in H\}$$ with $$(g_1,h_1)(g_2,h_2)=(g_1{g}_2,h_1{h}_2)\text{.}$$

$\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.=\{0,1\}$ and $$\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\times \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.=\{(0,0),(0,1),(1,0),(1,1)\}$$ with $$\begin{array}{ccccc}& (0,0)& (0,1)& (1,0)& (1,1)\\ (0,0)& (0,0)& (0,1)& (1,0)& (1,1)\\ (0,1)& (0,1)& (0,0)& (1,1)& (1,0)\\ (1,0)& (1,0)& (1,1)& (0,0)& (0,1)\\ (1,1)& (1,1)& (1,0)& (0,1)& (0,0)\end{array}\begin{array}{c}\text{Order of}\hspace{0.17em}\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\times \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\hspace{0.17em}\text{is}\hspace{0.17em}4\\ \text{Order of}\hspace{0.17em}(0,0)\hspace{0.17em}\text{is}\hspace{0.17em}1\\ \text{Order of}\hspace{0.17em}(1,0)\hspace{0.17em}\text{is}\hspace{0.17em}2\\ \text{Order of}\hspace{0.17em}(0,1)\hspace{0.17em}\text{is}\hspace{0.17em}2\\ \text{Order of}\hspace{0.17em}(1,1)\hspace{0.17em}\text{is}\hspace{0.17em}2\end{array}$$

$\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}$ is a subgroup of $S_4\text{.}$ $$\begin{array}{ccccc}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\end{array}\begin{array}{c}\text{Order of}\hspace{0.17em}\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\hspace{0.17em}\text{is}\hspace{0.17em}4\\ \text{Order of}\hspace{0.17em}\begin{array}{c}\end{array}\hspace{0.17em}\text{is}\hspace{0.17em}1\\ \text{Order of}\hspace{0.17em}\begin{array}{c}\end{array}\hspace{0.17em}\text{is}\hspace{0.17em}2\\ \text{Order of}\hspace{0.17em}\begin{array}{c}\end{array}\hspace{0.17em}\text{is}\hspace{0.17em}2\\ \text{Order of}\hspace{0.17em}\begin{array}{c}\end{array}\hspace{0.17em}\text{is}\hspace{0.17em}2\end{array}$$ and the function $$\begin{array}{ccc}\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\times \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.& ⟶& \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\\ (0,0)& ⟼& \begin{array}{c}\end{array}\\ (1,0)& ⟼& \begin{array}{c}\end{array}\\ (0,1)& ⟼& \begin{array}{c}\end{array}\\ (1,1)& ⟼& \begin{array}{c}\end{array}\end{array}$$ is an isomorphism.

Subgroups of $\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\times \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.\text{:}$ $$\begin{array}{c} ℤ2ℤ×ℤ2ℤ {(0,0),(1,0)} {(0,0),(0,1)} {(0,0),(1,1)} {(0,0)} \end{array}$$ Subgroups of $\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}:$ $$\begin{array}{c} {,,,} {,} {,} {,} {} \end{array}$$

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.

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