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 by left multiplication.
• acts on itself by conjunction.

Proposition

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

Proof Given an action defineed, 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.

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 .

Classification of Actions

Transitive Action

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

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 .

Proposition

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

The Orbit-Stabilizer Theorem

Conjugacy Classes

When acts on itself by conjugation, the orbits are called conjugacy classes.

Centraliser

The centralizer of an element in a group , is the set of elements that commute with . Or equivalently, the stabilizer of under conjugation is the centralizer of .

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:

The Class Equation

Let be a finite group. Then

Proof acts on itself by conjugation. It follows that is a disjoint union of conjugacy classes. For every class we have as by Orbit-Stabilizer Theorem we have .

-group

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

Corollary

Every -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 .

Lemma

For any group , is cyclic iff is abelian.

Cauchy’s Theorem

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

Proof