# Abstract Algebra/Group Theory/Group actions on sets

Interesting in it's own right, group actions are a useful tool in algebra and will permit us to prove the Sylow theorems, which in turn will give us a toolkit to describe certain groups in greater detail.

## Basics

[edit | edit source]**Definition 1.8.1**:

Let be an *arbitrary* set, and let be a group. A function

is called **group action** by on if and only if ( denoting the identity of )

- and
- .

When a certain group action is given in a context, we follow the prevalent convention to write simply for . In this notation, the requirements for a group action translate into

- and
- .

There is a one-to-one correspondence between group actions of on and homomorphisms .

**Definition 1.8.2**:

Let be a group and a set. Given a homomorphism , we may define a corresponding group action by

- .

If we are given a group action , then

is a homomorphism. The thus defined correspondence between homomorphisms and group actions is a bijective one.

**Proof**:

1.

Indeed, if is a homomorphism, then

- and
- .

2.

is bijective for all , since

- .

Let also . Then

- .

3.

We note that the constructions treated here are inverse to each other; indeed, if we transform a homomorphism to an action via

and then turn this into a homomorphism via

- ,

we note that since .

On the other hand, if we start with a group action , turn that into a homomorphism

and turn that back into a group action

- ,

then we ended up with the same group action as in the beginning due to .

**Examples 1.8.3**:

- acts on via .
- acts on via matrix multiplication: , where the first juxtaposition stands for the group action definition and the second for matrix multiplication.

## Types of actions

[edit | edit source]**Definitions 1.8.4**:

A group action is called

**faithful**iff ('identity on all elements of enforces identity on ')**free**iff ('different group elements map an to different elements of '), and**transitive**iff for all there exists such that .

Subtle analogies to real life become apparent if we note that an action is faithful if and only if for two distinct there exist such that , and it is free if and only if the elements are all different for all .

**Theorem 1.8.5**:

A free operation on a nonempty set is faithful.

**Proof**: .

We now attempt to characterise these three definitions; i.e. we try to find conditions equivalent to each.

**Theorem 1.8.6**:

A group action is faithful if and only if the induced homomorphism is injective.

**Proof**:

Let first a faithful action be given. Assume . Then for all and hence . Let now be injective. Then .

An important consequence is the following

**Corollary 1.8.7 (Cayley)**:

Every group is isomorphic to some subgroup of a symmetric group.

**Proof**:

A group acts on itself faithfully via left multiplication. Hence, by the previous theorem, there is a monomorphism .

For the characterisation of the other two definitions, we need more terminology.

## Orbit and stabilizer

[edit | edit source]**Definitions 1.8.8**:

Let be a group action, and let . Then

- is called the
**orbit of**and - is called the
**stabilizer of**. More generally, for a subset we define as the**stabilizer of**.

Using this terminology, we obtain a new characterisation of free operations.

**Theorem 1.8.9**:

An operation is free if and only if is trivial for each .

**Proof**: Let the operation be free and let . Then

- .

Since the operation is free, .

Assume that for each , is trivial, and let such that . The latter is equivalent to . Hence .

We also have a new characterisation of transitive operations using the orbit:

**Theorem 1.8.10**:

An operation is transitive if and only if for all .

**Proof**:

Assume for all , and let . Since transitivity follows.

Assume transitivity, and let . Then for all there exists with and hence .

Regarding the stabilizers we have the following two theorems:

**Theorem 1.8.11**:

Let be a group action and . Then .

**Proof**:

First of all, . Let . Then and hence . Further and hence .

**Theorem 1.8.12**:

Let . If we write for each , then

- .

**Proof**:

## Cardinality formulas

[edit | edit source]The following theorem will imply formulas for the cardinalities of , , or respectively.

**Theorem 1.8.13**:

Let an action be given. The relation is an equivalence relation, whose equivalence classes are given by the orbits of the action. Furthermore, for each the function

is a well-defined, bijective function.

**Proof**:

1.

- Reflexiveness:
- Symmetry:
- Transitivity: .

2.

Let be the equivalence class of . Then

- .

3.

Let . Since , . Hence, . Hence well-definedness. Surjectivity follows from the definition. Let . Then and thus . Hence injectivity.

**Corollary 1.8.14 (the orbit-stabilizer theorem)**:

Let an action be given, and let . Then

- , or equivalently .

**Proof**: By the previous theorem, the function is a bijection. Hence, . Further, by Lagrange's theorem .

**Corollary 1.8.15 (the orbit equation)**:

Let an action be given, and let be a complete and unambiguous list of the orbits. Then

- .

**Proof**: The first equation follows immediately from the equivalence classes of the relation from theorem 1.8.13 partitioning , and the second follows from Corollary 1.8.14.

**Corollary 1.8.16**:

Let an action be given, let , and let be a complete and unabiguous list of all nontrivial orbits (where the orbit of is said to be trivial iff ). Then

- .

**Proof**: This follows from the previous Corollary and the fact that equals the sum of the cardinalities the trivial orbits.

The following lemma, which is commonly known as *Burnside's lemma*, is actually due to Cauchy:

**Corollary 1.8.17 (Cauchy's lemma)**:

Let an action be given, where are finite. For each , we denote .

## The class equation

[edit | edit source]**Definition 1.8.18**:

Let a group act on itself by conjugation, i. e. for all . For each , the **centraliser** of is defined to be the set

- .

Using the machinery we developed above, we may now set up a formula for the cardinality of . In order to do so, we need a preliminary lemma though.

**Lemma 1.8.19**:

Let act on itself by conjugation, and let . Then the orbit of is trivial if and only if .

**Proof**: .

**Corollary 1.8.20 (the class equation)**:

Let be a group acting on itself by conjugation, and let be a complete and unambiguous list of the non-trivial orbits of that action. Then

- .

**Proof**: This follows from lemma 1.8.19 and Corollary 1.8.16.

## Special topics

[edit | edit source]### Equivariant functions

[edit | edit source]A set together with a group acting on it is an algebraic structure. Hence, we may define some sort of morphisms for those structures.

**Definition 1.8.21**:

Let a group act on the sets and . A function is called **equivariant** iff

- .

**Lemma 1.8.22**:

### p-groups

[edit | edit source]We shall now study the following thing:

**Definition 1.8.24**:

Let be a prime number. If is a group such that for some , then is called a **-group**.

**Corollary 23:** Let be a -group acting on a set . Then .

*Proof*: Since is a -group, divides for each with defined as in Lemma 21. Thus . ∎

### Group Representations

[edit | edit source]Linear group actions on vector spaces are especially interesting. These have a special name and comprise a subfield of group theory on their own, called *group representation theory*. We will only touch slightly upon it here.

**Definition 24:** Let be a group and be a vector space over a field . Then a *representation* of on is a map such that

- i) given by , , is linear in over .

- ii)

- iii) for all , .

V is called the *representation space* and the dimension of , if it is finite, is called the *dimension* or *degree* of the representation.

**Remark 25:** Equivalently, a representation of on is a homomorphism . A representation can be given by listing and , .

As a representation is a special kind of group action, all the concepts we have introduced for actions apply for representations.

**Definition 26:** A representation of a group on a vector space is called *faithful* or *effective* if is injective.