Group Actions

Group Action

Let be a group and a set. An action of on is a function

satisfying

• for all , and
• for all and

A set equipped with an action of is called a -set.

e.g.

• (the group) acts on (the set) by left multiplication.
• Let . Then acts on the left coset space by left multiplication.
• acts on itself by conjunction .
  

Proposition

A group action of on a set is the same as a homomorphism .

Proof Given an action defined, for every , a map by . Then is a bijection because is its inverse. Therefore, we get a map Now check that is a homomorphism. Indeed, Conversely, given such , define by . This is a group action and the two constructions we defined are inverses to each other.

Proposition

An action of a group on a set defines an equivalence relation on :

Proof Reflexivity follows from the fact that . Symmetry follows from the fact that if then . Transitivity follows from the fact that if and then .

Orbit and Stabiliser

Let be a group acting on a set and let . Then, the orbit of is the subset .
The stabilizer of in is the subgroup .

e.g. When acts on itself by conjugation, the orbits are called conjugacy classes, and the stabilizers are centralisers.

Classification of Actions

Transitive Action

A group action is transitive if it has only one orbit. This means for any two elements , there exists an element such that .

e.g. Let , then acts transitively on the left coset space by left multiplication, because any two cosets and are related by .

Free Action

A group action of on is said to be free if implies for all .

Faithful (Effective) Action

An action of on is said to be faithful if the only element of that acts as the identity on is the identity element of .

The Orbit-Stabilizer Theorem

Orbit-Stabilizer Theorem

Let be a group acting on a set . Let and let be the stabilizer of in and let ) be the orbit. Then the the map is a bijection. In particular, if is finite then . So the size of every orbit divides the order of the group.

Proof The map is surjective: an inverse image of is . The map is injective since: $$ gx = hx \iff h^{-1}gx=x \iff h^{-1}g\in G_x \iff gG_{x}=hG_{x}. $$$\square$

Class Equation

Let be a finite group, acting on a finite set . Then

Proof We claim that is a disjoint union of orbits. That is, and are either the same or disjoint. Suppose is nonempty and we pick some , then for some . It follows that , so . Similarly, . Hence . For every orbit we have by Orbit-Stabilizer Theorem.

e.g. If a finite group acts on itself by conjugation, then the orbits are the conjugacy classes and the stabilizers are the centralisers. The class equation is where are representatives of the non-trivial conjugacy classes.

-groups

-group

A finite group is called a -group if its order is a -power for some prime .

Corollary

Every $p$-group has a non-trivial centre.

Proof Let be a finite −group and make it act on itself by conjugation. Observe that:By class equation, we know that . Note that each is a -power, if were trivial, then there is only one term , and all other terms are at least , so for some , which is not divisible by , a contradiction. Therefore, is non-trivial.

Cauchy’s Theorem

If is a prime dividing the order of a finite group then contains an element of order .

Proof Let . The group acts on by This is a well-defined group action because the multiplication of the RHS is Observe that is a fixed point (i.e., ) if and only if and . If there were no elements of order , then only , and all other will have . It follows that for some . This yields a contradiction because must divides .

Semidirect Products

Semidirect Product

Given groups and , and an -action on by automorphisms, say , the semidirect product is the set , with multiplication defined by:

e.g. If is the trivial action then this construction reduces to the usual direct product .