NoteNextra-origin/content/Math4302/Math4302_L21.md

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.

Example

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.

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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.

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

Examples of isotropy subgroups

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\}.

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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.

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|}

Proof

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.