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 (unknown function "\lefttorightarrow"): {\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 $D_4=\{{\rho }_0,{\rho }_1,{\rho }_2,{\rho }_4,{\mu }_1,{\mu }_2,{\delta }_1,{\delta }_2\}$, 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 (unknown function "\lefttorightarrow"): {\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 $\{1,2,3,4\}$ $G_1$ is isotropy $\{{\rho }_0,{\rho }_2\}$ $\left|G\right|=8$

Applications to Counting

• Isotropy group: $G_x=\{g\in G\mid gx=x\}$
• Fixed Points: $X_g=(\mathrm{F}\mathrm{i}\mathrm{x}(g))=\{x\in X\mid gx=x\}$

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

$r\cdot \left|G\right|=\sum _{g\in G}\left|X_g\right|⟹r=\frac{1}{\left|G\right|}\sum _{g\in G}\left|X_g\right|$

(⇒) $W=\{(g,x)\mid gx=x\}$

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

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

(⇒)

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

$W=\underset{x\in X}{⨆}{G}_x$

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

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

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