# 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 .

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 .

## 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.