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: 25 September 2014

Lecture 31: Revision: Analogies

Let $G$ be a finite subgroup of $\text{Isom}\left({𝔼}^2\right)\text{.}$ Then $G$ is a cyclic or a dihedral group.

		Proof.
	Step 1 Let $p\in {𝔼}^2$ and $G=\{g_1,\dots ,g_t\}\text{.}$ Then $$q=g_{1}p+\cdots +g_{r}p$$ is a fixed point of $G\text{.}$ So every element of $G$ is a rotation about $q$ or a reflection in a line through $q\text{.}$ Let $h=r_{\theta ,q}$ with $\theta$ minimum possible and let $H=⟨h⟩\text{.}$ Then $H$ is a cyclic group. Case 1: If $H=G$ then $G$ is cyclic. Case 2: If $H\ne G$ let $s_1,s_2\in G$ such that $s_2,s_1\notin H$ and $s_1\ne s_2$ then $s_1{s}_2\in H$ so $s_1\in H{s}_2^{-1}=H{s}_2,$ since $s_2^2=1\text{.}$ So $G=\{H,s_{1}H\}$ with $s_1^2=1\text{.}$ So $G$ is a dihedral group. $\square$

Groups

A group is a set $G$ with a function $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}$$ such that

(a)	If $g_1,g_2,g_3\in G$ then $\left(g_1{g}_2\right)g_3=g_1\left(g_2{g}_3\right),$
(b)	There exists $1\in G$ such that if $g\in G$ then $g·1=g$ and $1·g=g\text{.}$
(c)	If $g\in G$ then there exists $g^{-1}\in G$ such that $g{g}^{-1}=1$ and $g^{-1}g=1\text{.}$

Homomorphisms are for comparing groups.

A homomorphism from $G$ to $H$ is a function $f:G\to H$ such that

(a)	If $g_1,g_2\in G$ then $f\left(g_1{g}_2\right)=f\left(g_1\right)f\left(g_2\right)\text{.}$

Let $f:G\to H$ be a homomorphism. The kernel of $f$ is $$\text{ker}\hspace{0.17em}f=\{g\in G\hspace{0.17em}|\hspace{0.17em}f(g=1)\}$$ and the image of $f$ is $$\text{im}\hspace{0.17em}f=\left\{f\left(g\right)\hspace{0.17em}\right|\hspace{0.17em}g\in G\}\text{.}$$

Examples of groups

Cyclic groups, Dihedral groups, Symmetric groups. $$\begin{array}{rcl}{\displaystyle S_n}& =& {\displaystyle \{\sigma :\{1,\dots ,n\}\to \{1,\dots ,n\}\hspace{0.17em}|\hspace{0.17em}\sigma \hspace{0.17em}\text{is a bijection}\}}\\ & =& {\displaystyle \genfrac{}{}{0ex}{}{{\displaystyle \genfrac{}{}{0ex}{}{\text{graphs with}\hspace{0.17em}n\hspace{0.17em}\text{top vertices and}\hspace{0.17em}n\hspace{0.17em}\text{bottom vertices}}{\text{such that each top dot is connected to}}}}{{\displaystyle \genfrac{}{}{0ex}{}{\text{exactly one bottom dot and each bottom dot}}{\text{is connected to exactly one top dot}}}}}\end{array}$$ with product given by composition ${\sigma }_1{\sigma }_2=\begin{array}{c} σ1 σ2 \end{array}$ $$\begin{array}{rcl}{\displaystyle S_3}& =& {\displaystyle \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\},}\\ {\displaystyle S_2}& =& {\displaystyle \{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\},}\\ {\displaystyle S_1}& =& {\displaystyle \left\{\begin{array}{c}\end{array}\right\}}\end{array}$$ and $$\begin{array}{ccccccc}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& & \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& & & \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& & & & \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}& \begin{array}{c}\end{array}& & & & \begin{array}{c}\end{array}& \begin{array}{c}\end{array}\end{array}$$ Note that $$\begin{array}{rcl}{\displaystyle A}& =& {\displaystyle \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\text{is a subgroup of}\hspace{0.17em}{S}_4,}\\ {\displaystyle A}& =& {\displaystyle ⟨\begin{array}{c}\end{array}⟩,\text{the group generated by}\hspace{0.17em}\begin{array}{c}\end{array}\text{.}}\end{array}$$ Also $$\begin{array}{rcl}{\displaystyle B}& =& {\displaystyle ⟨\begin{array}{c}\end{array},\begin{array}{c}\end{array}⟩}\\ & =& {\displaystyle \{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}}\end{array}$$ is a subgroup of $S_4\text{.}$

Notes and References

These are a typed copy of Lecture 31 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 18, 2011.

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