MATH 415 Lecture 25

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Actions on a Set

Example

Failed to parse (unknown function "\lefttorightarrow"): {\displaystyle G \lefttorightarrow G} by left ($L_{g}$) or right ($R_{h}$) multiplication.

$\sigma :G\times X\to X$ now becomes $\sigma :G\times G\to G$.

Observe that left and right multiplication commute:

$L_{g}\circ R_{h}=R_{h}\circ L_{g}$.

Example

Vector space $V$ and fields of scalars $F$, $\mathbb {R}$, or $\mathbb {C}$.

Then $V$ is a ($\mathbb {R} ^{*}$, $\mathbb {C} ^{*}$, or $F^{*}$)-set, where $F*$ is the multiplicative group of nonzero elements of a field.

Example

Let $Y$ be the set of cosets of $G$ formed from $H<G$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \lefttorightarrow Y} with $g_{1}(gH)=g_{1}gH$

If Failed to parse (unknown function "\lefttorightarrow"): {\displaystyle G \lefttorightarrow X} , then there is a bijection $\psi :X\to Y$ such that $\psi (gx)=g\psi (x)$

Example

Let $G$ be the dihedral group $D_{4}=\left\{\rho _{0},\rho _{1},\rho _{2},\rho _{4},\mu _{1},\mu _{2},\delta _{1},\delta _{2}\right\}$, where

• $\rho _{i}$ is a rotation by ${\frac {\pi \,i}{2}}$ radians (in which case $\rho _{0}$ is the identity element)
• $\mu _{1}$ and $\mu _{2}$ are mirroring on the vertical and horizontal axes, respectively
• $\delta _{1}$ and $\delta _{2}$ are mirroring along the NW-SE and NE-SW diagonals, respectively

Define set $X={1,2,3,4,s_{1},s_{2},s_{3},s_{4},m_{1},m_{2},d_{1},d_{2},c,p_{1},p_{2},p_{3},p_{4}}$ of parts of the dihedral square:

• $1$ through $4$ are vertices
• $s_{1}$ through $s_{4}$ are sides
• $m_{1}$ and $m_{2}$ are vertical and horizontal axes, respectively
• $d_{1}$ and $d_{2}$ are the NW-SE and NE-SW diagonals, respectively
• $c$ is the center point
• $p_{1}$ through $p_{4}$ are midpoints on each side

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_4 \lefttorightarrow X} .

We can fill out the operation table:

	$1$	$2$	$3$	$4$	$s_{1}$	$s_{2}$	$s_{3}$	$s_{4}$	$m_{1}$	$m_{2}$	$d_{1}$	$d_{2}$	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$
$\rho _{0}$
$\rho _{1}$
$\rho _{2}$
$\rho _{3}$
$\mu _{1}$
$\mu _{2}$
$\delta _{1}$
$\delta _{2}$

$G1$ is the orbit $\left\{1,2,3,4\right\}$ $G_{1}$ is isotropy $\left\{\rho _{0},\rho _{2}\right\}$ $\left|G\right|=8$

Applications to Counting

• Isotropy group: $G_{x}=\left\{g\in G~\mid ~g\,x=x\right\}$
• Fixed Points: $X_{g}=(\mathrm {Fix} (g))=\left\{x\in X~\mid ~g\,x=x\right\}$

Burnside Formula

Theorem 17.1. Let $G$ be a finite group and $X$ is a finite $G$-set. If $r$ is a number of orbits in $X$ under $G$, then

$r\cdot \left|G\right|=\sum _{g\in G}\left|X_{g}\right|\quad \implies \quad r={\frac {1}{\left|G\right|}}\sum _{g\in G}\left|X_{g}\right|$

(⇒) $W=\left\{\left(g,x\right)~\mid ~g\,x=x\right\}$

$N=\left|W\right|$ for each $g$.

$W$ is a disjoint union of $X_{g}$ for each $g$, then $N=\sum _{g\in G}\left|X_{g}\right|$

(⇒)

$|GX|=(G:G_{x})={\frac {|G|}{|G_{x}|}}$

$W=\bigsqcup _{x\in X}G_{x}$

$N=\sum _{x\in X}\left|G_{x}\right|=\sum _{x\in X}{\frac {\left|G\right|}{\left|Gx\right|}}=\left|G\right|\,\sum _{x\in X}{\frac {1}{\left|Gx\right|}}$

$\sum _{y\in Gx}{\frac {1}{\left|Gx\right|}}={\frac {1}{\left|Gx\right|}}\sum _{y\in Gx}1={\frac {\left|Gx\right|}{\left|Gx\right|}}=1$

$N=\left|G\right|({\mbox{number of orbits in}}\ X\ {\mbox{under}}\ G)=|G|\cdot r$