# 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

Definition 1.8.1:

Let ${\displaystyle X}$  be an arbitrary set, and let ${\displaystyle G}$  be a group. A function

${\displaystyle f:G\times X\to X}$

is called group action by ${\displaystyle G}$  on ${\displaystyle X}$  if and only if (${\displaystyle \iota }$  denoting the identity of ${\displaystyle G}$ )

1. ${\displaystyle \forall x\in X:f(\iota ,x)=x}$  and
2. ${\displaystyle \forall \sigma ,\tau \in G,x\in X:f(\sigma ,f(\tau ,x))=f(\sigma \tau ,x)}$ .

When a certain group action is given in a context, we follow the prevalent convention to write simply ${\displaystyle \sigma x}$  for ${\displaystyle f(\sigma ,x)}$ . In this notation, the requirements for a group action translate into

1. ${\displaystyle \forall x\in X:\iota x=x}$  and
2. ${\displaystyle \forall \sigma ,\tau \in G,x\in X:\sigma (\tau x)=(\sigma \tau )x}$ .

There is a one-to-one correspondence between group actions of ${\displaystyle G}$  on ${\displaystyle X}$  and homomorphisms ${\displaystyle G\to S_{X}}$ .

Definition 1.8.2:

Let ${\displaystyle G}$  be a group and ${\displaystyle X}$  a set. Given a homomorphism ${\displaystyle \varphi :G\to S_{X}}$ , we may define a corresponding group action by

${\displaystyle \sigma x:=\varphi (\sigma )(x)}$ .

If we are given a group action ${\displaystyle G\times X\to X}$ , then

${\displaystyle \varphi (\sigma ):=x\mapsto \sigma x}$

is a homomorphism. The thus defined correspondence between homomorphisms ${\displaystyle G\to S_{X}}$  and group actions ${\displaystyle G\times X\to X}$  is a bijective one.

Proof:

1.

Indeed, if ${\displaystyle \varphi :G\to S_{X}}$  is a homomorphism, then

${\displaystyle \iota x=\varphi (\iota )(x)={\text{Id}}(x)=x}$  and
${\displaystyle \sigma (\tau x)=\varphi (\sigma )(\varphi (\tau )(x))=(\varphi (\sigma )\circ \varphi (\tau ))(x)=(\sigma \tau )(x)}$ .

2.

${\displaystyle \varphi (\sigma )}$  is bijective for all ${\displaystyle \sigma \in G}$ , since

${\displaystyle \varphi (\sigma )(x)=\varphi (\sigma )(y)\Leftrightarrow \sigma x=\sigma y\Leftrightarrow \sigma ^{-1}\sigma x=\sigma ^{-1}\sigma y}$ .

Let also ${\displaystyle \tau \in G}$ . Then

${\displaystyle \varphi (\sigma \tau )=x\mapsto (\sigma \tau )x=x\mapsto \sigma (\tau (x))=(x\mapsto \sigma x)\circ (x\mapsto \tau x)=\varphi (\sigma )\circ \varphi (\tau )}$ .

3.

We note that the constructions treated here are inverse to each other; indeed, if we transform a homomorphism ${\displaystyle \varphi :G\to S_{X}}$  to an action via

${\displaystyle \sigma x:=\varphi (\sigma )(x)}$

and then turn this into a homomorphism via

${\displaystyle \psi :G\to S_{X},\psi (\sigma ):=x\mapsto \sigma x}$ ,

we note that ${\displaystyle \psi =\phi }$  since ${\displaystyle \psi (\sigma )=x\mapsto \sigma x=x\mapsto \varphi (\sigma )(x)=\varphi (\sigma )}$ .

On the other hand, if we start with a group action ${\displaystyle G\times X\to X}$ , turn that into a homomorphism

${\displaystyle \varphi (\sigma ):=x\mapsto \sigma x}$

and turn that back into a group action

${\displaystyle \sigma x:=\varphi (\sigma )(x)}$ ,

then we ended up with the same group action as in the beginning due to ${\displaystyle \varphi (\sigma )(x)=\sigma x}$ .${\displaystyle \Box }$

Examples 1.8.3:

1. ${\displaystyle S_{n}}$  acts on ${\displaystyle \mathbb {R} ^{n}}$  via ${\displaystyle \sigma (x_{1},\ldots ,x_{n})=(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}$ .
2. ${\displaystyle GL_{n}(\mathbb {R} )}$  acts on ${\displaystyle \mathbb {R} ^{n}}$  via matrix multiplication: ${\displaystyle Ax:=Ax}$ , where the first juxtaposition stands for the group action definition and the second for matrix multiplication.

## Types of actions

Definitions 1.8.4:

A group action ${\displaystyle G\times X\to X}$  is called

1. faithful iff ${\displaystyle (\forall x\in X:\sigma x=\tau x)\Rightarrow \sigma =\tau }$  ('identity on all elements of ${\displaystyle x}$  enforces identity on ${\displaystyle G}$ ')
2. free iff ${\displaystyle (\exists x\in X:\sigma x=\tau x)\Rightarrow \sigma =\tau }$  ('different group elements map an ${\displaystyle x}$  to different elements of ${\displaystyle X}$ '), and
3. transitive iff for all ${\displaystyle x,y\in X}$  there exists ${\displaystyle \sigma \in G}$  such that ${\displaystyle y=\sigma x}$ .

Subtle analogies to real life become apparent if we note that an action is faithful if and only if for two distinct ${\displaystyle \sigma \neq \tau \in G}$  there exist ${\displaystyle x\in X}$  such that ${\displaystyle \sigma x\neq \tau x}$ , and it is free if and only if the elements ${\displaystyle \sigma x,\sigma \in G}$  are all different for all ${\displaystyle x\in X}$ .

Theorem 1.8.5:

A free operation on a nonempty set is faithful.

Proof: ${\displaystyle (\forall x\in X:\sigma x=\tau x)\Rightarrow (\exists x\in X:\sigma x=\tau x)\Rightarrow \sigma =\tau }$ .${\displaystyle \Box }$

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 ${\displaystyle G\times X\to X}$  is faithful if and only if the induced homomorphism ${\displaystyle \varphi :G\to S_{X}}$  is injective.

Proof:

Let first a faithful action ${\displaystyle G\times X\to X}$  be given. Assume ${\displaystyle \varphi (\sigma )=\varphi (\tau )}$ . Then for all ${\displaystyle x\in X}$  ${\displaystyle \sigma x=\varphi (\sigma )(x)=\varphi (\tau )(x)=\tau x}$  and hence ${\displaystyle \sigma =\tau }$ . Let now ${\displaystyle \varphi }$  be injective. Then ${\displaystyle }$.

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 ${\displaystyle G\to S_{G}}$ .${\displaystyle \Box }$

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

## Orbit and stabilizer

Definitions 1.8.8:

Let ${\displaystyle G\times X\to X}$  be a group action, and let ${\displaystyle x\in X}$ . Then

• ${\displaystyle G(x):=\{\sigma x|\sigma \in G\}}$  is called the orbit of ${\displaystyle x}$  and
• ${\displaystyle G_{x}:=\{\sigma \in G|\sigma x=x\}}$  is called the stabilizer of ${\displaystyle x}$ . More generally, for a subset ${\displaystyle Y\subseteq X}$  we define ${\displaystyle G_{Y}:=\{\sigma \in G|\forall y\in Y:\sigma y\in Y\}}$  as the stabilizer of ${\displaystyle Y}$ .

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

Theorem 1.8.9:

An operation ${\displaystyle G\times X\to X}$  is free if and only if ${\displaystyle G_{x}}$  is trivial for each ${\displaystyle x\in X}$ .

Proof: Let the operation be free and let ${\displaystyle x\in X}$ . Then

${\displaystyle \sigma \in G_{x}\Leftrightarrow \sigma x=x=\iota x}$ .

Since the operation is free, ${\displaystyle \sigma =\iota }$ .

Assume that for each ${\displaystyle x\in X}$ , ${\displaystyle G_{x}}$  is trivial, and let ${\displaystyle y\in X}$  such that ${\displaystyle \sigma y=\tau y}$ . The latter is equivalent to ${\displaystyle \tau ^{-1}\sigma y=y}$ . Hence ${\displaystyle \tau ^{-1}\sigma \in G_{y}=\{\iota \}}$ .${\displaystyle \Box }$

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

Theorem 1.8.10:

An operation ${\displaystyle G\times X\to X}$  is transitive if and only if for all ${\displaystyle x\in X}$  ${\displaystyle G(x)=X}$ .

Proof:

Assume for all ${\displaystyle x\in X}$  ${\displaystyle G(x)=X}$ , and let ${\displaystyle y,z\in X}$ . Since ${\displaystyle G(y)=X\ni z}$  transitivity follows.

Assume transitivity, and let ${\displaystyle x\in X}$ . Then for all ${\displaystyle y\in X}$  there exists ${\displaystyle \sigma \in G}$  with ${\displaystyle \sigma x=y}$  and hence ${\displaystyle y\in G(x)}$ .${\displaystyle \Box }$

Regarding the stabilizers we have the following two theorems:

Theorem 1.8.11:

Let ${\displaystyle G\times X\to X}$  be a group action and ${\displaystyle x\in X}$ . Then ${\displaystyle G_{x}\leq G}$ .

Proof:

First of all, ${\displaystyle \iota \in G_{x}}$ . Let ${\displaystyle \sigma ,\tau \in G_{x}}$ . Then ${\displaystyle (\sigma \tau )x=\sigma (\tau x)=\sigma x=x}$  and hence ${\displaystyle \sigma \tau \in G_{x}}$ . Further ${\displaystyle \sigma ^{-1}x=\sigma ^{-1}\sigma x=x}$  and hence ${\displaystyle \sigma ^{-1}\in G_{x}}$ .${\displaystyle \Box }$

Theorem 1.8.12:

Let ${\displaystyle Y\subseteq X}$ . If we write ${\displaystyle \sigma Y:=\{\sigma y|y\in Y\}}$  for each ${\displaystyle \sigma \in G}$ , then

${\displaystyle G_{\sigma Y}=\sigma G_{Y}\sigma ^{-1}}$ .

Proof:

{\displaystyle {\begin{aligned}\tau \in G_{\sigma Y}&\Leftrightarrow \tau \sigma Y=\sigma Y\\&\Leftrightarrow \sigma ^{-1}\tau \sigma Y=Y\\&\Leftrightarrow \sigma ^{-1}\tau \sigma Y\in G_{Y}\\&\Leftrightarrow \tau \in \sigma G_{Y}\sigma ^{-1}\end{aligned}}} ${\displaystyle \Box }$

## Cardinality formulas

The following theorem will imply formulas for the cardinalities of ${\displaystyle G_{x}}$ , ${\displaystyle |G|}$ , ${\displaystyle (G:G_{x})}$  or ${\displaystyle X}$  respectively.

Theorem 1.8.13:

Let an action ${\displaystyle G\times X\to X}$  be given. The relation ${\displaystyle x\sim y:\Leftrightarrow \exists \sigma \in G:\sigma x=y}$  is an equivalence relation, whose equivalence classes are given by the orbits of the action. Furthermore, for each ${\displaystyle x\in X}$  the function

${\displaystyle \{\sigma G_{x}|\sigma \in G\}\to G(x),\sigma G_{x}\mapsto \sigma x}$

is a well-defined, bijective function.

Proof:

1.

• Reflexiveness: ${\displaystyle \iota x=\iota }$
• Symmetry: ${\displaystyle \sigma x=y\Leftrightarrow x=\sigma ^{-1}y}$
• Transitivity: ${\displaystyle \sigma x=y\wedge \tau y=z\Rightarrow (\tau \sigma )x=z}$ .

2.

Let ${\displaystyle [x]}$  be the equivalence class of ${\displaystyle x}$ . Then

${\displaystyle y\in [x]\Leftrightarrow \exists \sigma \in G:\sigma x=y\Leftrightarrow y\in G(x)}$ .

3.

Let ${\displaystyle \sigma G_{x}=\tau G_{x}}$ . Since ${\displaystyle G_{x}\leq G}$ , ${\displaystyle \tau ^{-1}\sigma \in G_{x}}$ . Hence, ${\displaystyle \tau ^{-1}\sigma x=x\Leftrightarrow \tau x=\sigma x}$ . Hence well-definedness. Surjectivity follows from the definition. Let ${\displaystyle \sigma x=\tau x}$ . Then ${\displaystyle \tau ^{-1}\sigma x=x}$  and thus ${\displaystyle \tau ^{-1}\sigma G_{x}=G_{x}}$ . Hence injectivity.${\displaystyle \Box }$

Corollary 1.8.14 (the orbit-stabilizer theorem):

Let an action ${\displaystyle G\times X\to X}$  be given, and let ${\displaystyle x\in X}$ . Then

${\displaystyle |G(x)|=(G:G_{x})}$ , or equivalently ${\displaystyle |G(x)|\cdot |G_{x}|=|G|}$ .

Proof: By the previous theorem, the function ${\displaystyle \{\sigma G_{x}|\sigma \in G\}\to G(x),\sigma G_{x}\mapsto \sigma x}$  is a bijection. Hence, ${\displaystyle (G:g_{x})=|G(x)|}$ . Further, by Lagrange's theorem ${\displaystyle (G:G_{x})={\frac {|G|}{|G_{x}|}}}$ .${\displaystyle \Box }$

Corollary 1.8.15 (the orbit equation):

Let an action ${\displaystyle G\times X\to X}$  be given, and let ${\displaystyle G(x_{1}),\ldots ,G(x_{n})}$  be a complete and unambiguous list of the orbits. Then

${\displaystyle |X|=\sum _{j=1}^{n}\left|G\left(x_{j}\right)\right|=\sum _{j=1}^{n}(G:G_{x_{j}})}$ .

Proof: The first equation follows immediately from the equivalence classes of the relation from theorem 1.8.13 partitioning ${\displaystyle X}$ , and the second follows from Corollary 1.8.14.${\displaystyle \Box }$

Corollary 1.8.16:

Let an action ${\displaystyle G\times X\to X}$  be given, let ${\displaystyle Z=\{x\in X|\forall \sigma \in G:\sigma x=x\}}$ , and let ${\displaystyle G(x_{1}),\ldots ,G(x_{m})}$  be a complete and unabiguous list of all nontrivial orbits (where the orbit of ${\displaystyle x\in X}$  is said to be trivial iff ${\displaystyle G(x)=\{x\}}$ ). Then

${\displaystyle |X|=|Z|+\sum _{j=1}^{m}\left|G\left(x_{j}\right)\right|}$ .

Proof: This follows from the previous Corollary and the fact that ${\displaystyle |Z|}$  equals the sum of the cardinalities the trivial orbits.${\displaystyle \Box }$

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 ${\displaystyle G\times X\to X}$  be given, where ${\displaystyle G,X}$  are finite. For each ${\displaystyle \sigma \in G}$ , we denote ${\displaystyle }$.

## The class equation

Definition 1.8.18:

Let a group ${\displaystyle G}$  act on itself by conjugation, i. e. ${\displaystyle \sigma x:=\sigma x\sigma ^{-1}}$  for all ${\displaystyle \sigma ,x\in G}$ . For each ${\displaystyle x\in G}$ , the centraliser of ${\displaystyle x}$  is defined to be the set

${\displaystyle {\mathcal {C}}_{G}(x):=G_{x}}$ .

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

Lemma 1.8.19:

Let ${\displaystyle G}$  act on itself by conjugation, and let ${\displaystyle x\in G}$ . Then the orbit of ${\displaystyle x}$  is trivial if and only if ${\displaystyle x\in Z(G)}$ .

Proof: ${\displaystyle x\in Z(G)\Leftrightarrow \forall \sigma \in G:\sigma x\sigma ^{-1}=x\Leftrightarrow G(x)=\{x\}}$ .${\displaystyle \Box }$

Corollary 1.8.20 (the class equation):

Let ${\displaystyle G}$  be a group acting on itself by conjugation, and let ${\displaystyle G(x_{1}),\ldots ,G(x_{m})}$  be a complete and unambiguous list of the non-trivial orbits of that action. Then

${\displaystyle |G|=|Z(G)|+\sum _{j=1}^{m}(G:{\mathcal {C}}_{G}(x_{j}))}$ .

Proof: This follows from lemma 1.8.19 and Corollary 1.8.16.${\displaystyle \Box }$

## Special topics

### Equivariant functions

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 ${\displaystyle G}$  act on the sets ${\displaystyle X}$  and ${\displaystyle Y}$ . A function ${\displaystyle f:X\to Y}$  is called equivariant iff

${\displaystyle \forall \sigma \in G,x\in X:\sigma f(x)=f(\sigma x)}$ .

Lemma 1.8.22:

### p-groups

We shall now study the following thing:

Definition 1.8.24:

Let ${\displaystyle p}$  be a prime number. If ${\displaystyle G}$  is a group such that ${\displaystyle |G|=p^{k}}$  for some ${\displaystyle k\in \mathbb {N} }$ , then ${\displaystyle G}$  is called a ${\displaystyle p}$ -group.

Corollary 23: Let ${\displaystyle G}$  be a ${\displaystyle p}$ -group acting on a set ${\displaystyle S}$ . Then ${\displaystyle |S|\equiv |Z|\ \mathrm {mod} \ p}$ .

Proof: Since ${\displaystyle G}$  is a ${\displaystyle p}$ -group, ${\displaystyle p}$  divides ${\displaystyle |G*a|}$  for each ${\displaystyle a\in A}$  with ${\displaystyle A}$  defined as in Lemma 21. Thus ${\displaystyle \sum _{a\in A}|G*a|\equiv 0\ \mathrm {mod} \ p}$ .

### Group Representations

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 ${\displaystyle G}$  be a group and ${\displaystyle V}$  be a vector space over a field ${\displaystyle F}$ . Then a representation of ${\displaystyle G}$  on ${\displaystyle V}$  is a map ${\displaystyle \Phi \,:\,G\times V\rightarrow V}$  such that

i) ${\displaystyle \Phi (g)\,:\,V\rightarrow V}$  given by ${\displaystyle \Psi (g)(v)=\Psi (g,v)}$ , ${\displaystyle v\in V}$ , is linear in ${\displaystyle v}$  over ${\displaystyle F}$ .
ii) ${\displaystyle \Phi (e,v)=v}$
iii) ${\displaystyle \Phi \left(g_{1},\Phi (g_{2},v)\right)=\Phi (g_{1}g_{2},v)}$  for all ${\displaystyle g_{1},g_{2}\in G}$ , ${\displaystyle v\in V}$ .

V is called the representation space and the dimension of ${\displaystyle V}$ , if it is finite, is called the dimension or degree of the representation.

Remark 25: Equivalently, a representation of ${\displaystyle G}$  on ${\displaystyle V}$  is a homomorphism ${\displaystyle \phi \,:\,G\rightarrow GL(V,F)}$ . A representation can be given by listing ${\displaystyle V}$  and ${\displaystyle \phi }$ , ${\displaystyle (V,\phi )}$ .

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 ${\displaystyle G}$  on a vector space ${\displaystyle V}$  is called faithful or effective if ${\displaystyle \phi \,:\,G\rightarrow GL(V,F)}$  is injective.