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 25: Group actions, orbits, stabilizers

The dihedral group $D_n$ is the set $$D_n=\{1,x,x^2,\dots ,x^{n-1},y,xy,x^{2}y,\dots ,x^{n-1}y\}$$ with operation given by $$\left(x^i{y}^j\right)\left(x^k{y}^{\ell }\right)=x^{(i-k\hspace{0.17em}\text{mod}\hspace{0.17em}n)}{y}^{(j+\ell \hspace{0.17em}\text{mod}\hspace{0.17em}2)}$$ so that, in particular $$y^2=1,x^n=1\text{and}yx=x^{-1}y\text{.}$$

A $G\text{-set}$ $S,$ or an action of a group $G$ on a set $S$ is a function $$\begin{array}{ccc}G\times S& ⟶& S\\ (g,s)& ⟼& gs\end{array}$$ such that

(a)	If $g_1,g_2\in G$ and $s\in S$ then $g_1\left(g_{2}s\right)=\left(g_1{g}_2\right)s,$
(b)	If $s\in S$ then $1·s=s\text{.}$

Let $s\in S\text{.}$ The stabilizer of $s$ is $$G_s=\{g\in G\hspace{0.17em}|\hspace{0.17em}gs=s\}\text{.}$$ The orbit of $s$ is $$Gs=\left\{gs\hspace{0.17em}\right|\hspace{0.17em}g\in G\}\text{.}$$

Let $G$ be a group and let $S$ be a $G\text{-set.}$

(a)	The orbits of $G$ acting on $S$ partition $S\text{.}$
(b)	Let $s\in S\text{.}$ Then $G_s$ is a subgroup of $G,$ $$\begin{array}{cccc}\phi :& \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_s$}\right.& ⟶& Gs\\ & g{G}_s& ⟼& gs\end{array}$$ is a well defined function and $\phi$ is a bijection.

$\text{"}{D}_4$ acting on a square". $$\begin{array}{cccc}{}_b{}^a\begin{array}{c} e1 e2 e3 e4 \end{array}_c^d& {}_a{}^d\begin{array}{c}\end{array}_b^c& {}_d{}^c\begin{array}{c}\end{array}_a^b& {}_c{}^b\begin{array}{c}\end{array}_d^a\\ 1& x& x^2& x^3\\ {}_c{}^d\begin{array}{c}\end{array}_b^a& {}_d{}^a\begin{array}{c}\end{array}_c^b& {}_a{}^b\begin{array}{c}\end{array}_d^c& {}_b{}^c\begin{array}{c}\end{array}_a^d\\ y& xy& x^{2}y& x^{3}y\end{array}$$ $D_4$ acts on the vertices $V=\{a,b,c,d\}\text{:}$ $$xa=b,xya=a\text{.}$$ $D_4$ acts on the edges $E=\{e_1,e_2,e_3,e_4\}\text{:}$ $$x^2{e}_1=e_3,y{e}_1=e_1\text{.}$$

$\text{"}G$ acts on itself by conjugation". $$\begin{array}{ccc}G\times G& ⟶& G\\ (g,s)& ⟼& gs{g}^{-1}\end{array}\text{or}g·s=gs{g}^{-1},$$ where $g\in G$ and $s\in G\text{.}$

Let $s\in G\text{.}$ The centraliser of $s$ in $G$ is the stabiliser of $s,$ $${𝒵}_G\left(s\right)=\text{Stab}\left(s\right)=\{g\in G\hspace{0.17em}|\hspace{0.17em}gs{g}^{-1}=s\}\text{.}$$ The conjugacy class of $s$ in $G$ is the orbit of $s,$ $${𝒞}_s=\left\{gs{g}^{-1}\hspace{0.17em}\right|\hspace{0.17em}g\in G\}\text{.}$$ The centre of $G$ is $$𝒵\left(G\right)=\{s\in G\hspace{0.17em}|\hspace{0.17em}gs=sg\hspace{0.17em}\text{for all}\hspace{0.17em}g\in G\}\text{.}$$

HW: Show that $s\in 𝒵\left(G\right)$ if and only if ${𝒵}_G\left(s\right)=G\text{.}$

HW: Show that $s\in 𝒵\left(G\right)$ if and only if ${𝒞}_s=\left\{s\right\}\text{.}$

$D_4$ acts on itself by conjugation. $$x\begin{array}{c}↺\end{array}1\begin{array}{c}↻\end{array}y\text{and}x\begin{array}{c}↺\end{array}{x}^2\begin{array}{c}↻\end{array}y$$ $$x\begin{array}{c}↺\end{array}x\underset{y}{\overset{y}{\rightleftarrows }}{x}^3\begin{array}{c}↺\end{array}x\text{since}\begin{array}{ccc}{\displaystyle yx{y}^{-1}}& =& {\displaystyle x^{3}y{y}^{-1}=x^3}\\ {\displaystyle y{x}^2{y}^{-1}}& =& {\displaystyle x^{2}y{y}^{-1}=x^2}\\ {\displaystyle y{x}^3{y}^{-1}}& =& {\displaystyle xy{y}^{-1}=x}\end{array}$$ $$y\begin{array}{c}↺\end{array}y\underset{x}{\overset{x}{\rightleftarrows }}{x}^{2}y\begin{array}{c}↻\end{array}y\text{since}xy{x}^{-1}=xy{x}^3=xxy=x^{2}y$$ $$xy\underset{x,y}{\overset{y,x}{\rightleftarrows }}{x}^{3}y$$ So the conjugacy classes are $$\left\{1\right\},\left\{x^2\right\},\{x,x^3\},\{y,x^{2}y\},\{xy,x^{3}y\}$$ and the centralizers of elements in $D_4$ are $$\begin{array}{ccc}{\displaystyle {𝒵}_{D_4}\left(1\right)}& =& {\displaystyle \text{Stab}\left(1\right)=D_4}\\ {\displaystyle {𝒵}_{D_4}\left(x\right)}& =& {\displaystyle \text{Stab}\left(x\right)=\{1,x,x^2,x^3\}}\\ {\displaystyle {𝒵}_{D_4}\left(x^2\right)}& =& {\displaystyle \text{Stab}\left(x^2\right)=D_4}\\ {\displaystyle {𝒵}_{D_4}\left(x^3\right)}& =& {\displaystyle \text{Stab}\left(x^3\right)=\{1,x,x^2,x^3\}}\end{array}\begin{array}{ccc}{\displaystyle {𝒵}_{D_4}\left(y\right)}& =& {\displaystyle \text{Stab}\left(y\right)=\{1,x^2,y,x^{2}y\}}\\ {\displaystyle {𝒵}_{D_4}\left(xy\right)}& =& {\displaystyle \text{Stab}\left(xy\right)=\{1,x^2,xy,x^{3}y\}}\\ {\displaystyle {𝒵}_{D_4}\left(x^{2}y\right)}& =& {\displaystyle \text{Stab}\left(x^{2}y\right)=\{1,x^2,y,x^{2}y\}}\\ {\displaystyle {𝒵}_{D_4}\left(x^{3}y\right)}& =& {\displaystyle \text{Stab}\left(x^{3}y\right)=\{1,x^2,xy,x^{3}y\}}\end{array}$$

Notes and References

These are a typed copy of Lecture 25 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 4, 2011.

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