# Math4302 Modern Algebra (Lecture 21)

## Groups

### Group acting on a set

#### Definition of orbits

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 $Gx\cap Gy=\emptyset$.

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

<details>
<summary>Example</summary>

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 $(1,2)$:

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

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.

</details>

#### Definition of isotropy subgroup

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 $g^{-1}\in G_x$

<details>
<summary>Examples of isotropy subgroups</summary>

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.

</details>

#### Orbit-stabilizer theorem

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

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

<details>
<summary>Proof</summary>

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.

</details>