Math4302 Modern Algebra (Lecture 21)

Groups

We define the equivalence relation on $X$

x\sim y\iff y=g\cdot x\text{ for some }g

So we get a partition of $X$ into equivalence classes: orbits

Gx\coloneqq \{g\cdot x|g\in G\}=\{y\in X|x\sim y\}

is the orbit of $X$ .

$x,y\in X$ either $Gx=Gy$ or $Gx\cap Gy=\emptyset$ .

$X=\bigcup_{x\in X}Gx$ .

Let $D_4$ acting on $X=\{1,2,3,4\}$ . Let $D_4=\{e,\rho,\rho^2,\rho^3,\mu,\mu\rho,\mu\rho^2,\mu\rho^3\}$ .

define $\phi\in D_4$ , $i\in X$ , $\phi\cdot i=\phi(i)$

The orbits are:

orbit of 1: $D_4\cdot 1=\{1,2,3,4\}$ . This is equal to orbit of 2,3,4.

Let $G=S_3$ acting on $X=S_3$ via conjugation, let $\sigma\in X$ and $\phi\in G$ , we define $\phi\cdot\sigma\coloneqq \phi\sigma\phi^{-1}$ .

$S_3=\{e,(1,2,3),(1,3,2),(1,2),(1,3),(2,3)\}$ .

The orbits are:

orbit of $e$ : $G e=\{e\}$ . since $geg^{-1}=e$ for all $g\in S_3$ .

orbit of $(1,2,3)$ :

$e(1,2,3)e^{-1}=(1,2,3)$
$(1,3,2)(1,2,3)(1,3,2)^{-1}=(1,2,3)$
$(1,2,3)(1,2,3)(1,2,3)^{-1}=(1,2,3)$
$(1,2)(1,2,3)(1,2)^{-1}=(2,3)(1,2)=(1,3,2)$
$(1,3)(1,2,3)(1,3)^{-1}=(1,2)(1,3)=(1,3,2)$
$(2,3)(1,2,3)(2,3)^{-1}=(1,3)(2,3)=(1,3,2)$

So the orbit of $(1,2,3)$ is equal to orbit of $(1,3,2)$ . $=\{(1,2,3),(2,3,1)\}$ .

orbit of $(1,2)$ :

Therefore orbit of $(1,2)$ is equal to orbit of $(2,3)$ , $(1,3)$ . $=\{(1,2),(2,3),(1,3)\}$

The orbits may not have the same size.

Let $X$ be a $G$ -set, the stabilizer (or isotropy subgroup) corresponding to $x\in X$ is

G_x=\{g\in G|g\cdot x=x\}

$G_x$ is a subgroup of $G$ . $G_x\leq G$ .

$e\cdot x=x$ , so $e\in G_x$
If $g_1,g_2\in G_x$ , then $(g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)=g_1 \cdot x$ , so $g_1g_2\in G_x$
If $g\in G_x$ , then $g^{-1}\cdot g=x=g^{-1}\cdot x$ , so $g^{-1}\in G_x$

Let $D_4$ acting on $X=\{1,2,3,4\}$ , find $G_1$ , $G_2$ , $G_3$ , $G_4$ .

$G_1=G_3=\{e,\mu\}$ , $G_2=G_4=\{e,\mu\rho^2\}$ .

Let $S_3$ acting on $X=S_3$ . Find $G_{e}$ , $G_{(1,2,3)}$ , $G_{(1,2)}$ .

$G_{e}=S_3$ , $G_{(1,2,3)}=G_{(1,3,2)}=\{e,(1,2,3),(1,3,2)\}$ , $G_{(1,2)}=\{e,(1,2)\}$ , ( $G_{(1,3)}=\{e,(1,3)\}$ , $G_{(2,3)}=\{e,(2,3)\}$ )

The larger the orbit, the smaller the stabilizer.

If $X$ is a $G$ -set and $x\in X$ , then

|Gx|=(G:G_x)=\text{ number of left cosets of }G_x=\frac{|G|}{|G_x|}

Define $\alpha$ be the function that maps the set of left cosets of $G_x$ to orbit of $x$ . $gG_X\mapsto g\cdot x$ .

This function is well defined. And $\alpha$ is a bijection.

Continue next lecture.