# 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 $X$  be an arbitrary set, and let $G$  be a group. A function

$f:G\times X\to X$

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

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

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

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

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

Definition 1.8.2:

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

$\sigma x:=\varphi (\sigma )(x)$ .

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

$\varphi (\sigma ):=x\mapsto \sigma x$

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

Proof:

1.

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

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

2.

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

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

Let also $\tau \in G$ . Then

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

3.

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

$\sigma x:=\varphi (\sigma )(x)$

and then turn this into a homomorphism via

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

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

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

$\varphi (\sigma ):=x\mapsto \sigma x$

and turn that back into a group action

$\sigma x:=\varphi (\sigma )(x)$ ,

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

Examples 1.8.3:

1. $S_{n}$  acts on $\mathbb {R} ^{n}$  via $\sigma (x_{1},\ldots ,x_{n})=(x_{\sigma (1)},\ldots ,x_{\sigma (n)})$ .
2. $GL_{n}(\mathbb {R} )$  acts on $\mathbb {R} ^{n}$  via matrix multiplication: $Ax:=Ax$ , 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 $G\times X\to X$  is called

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

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

Theorem 1.8.5:

A free operation on a nonempty set is faithful.

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

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

Proof:

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

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

## Orbit and stabilizer

Definitions 1.8.8:

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

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

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

Theorem 1.8.9:

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

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

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

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

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

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

Theorem 1.8.10:

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

Proof:

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

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

Regarding the stabilizers we have the following two theorems:

Theorem 1.8.11:

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

Proof:

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

Theorem 1.8.12:

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

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

Proof:

{\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}} $\Box$

## Cardinality formulas

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

Theorem 1.8.13:

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

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

is a well-defined, bijective function.

Proof:

1.

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

2.

Let $[x]$  be the equivalence class of $x$ . Then

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

3.

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

Corollary 1.8.14 (the orbit-stabilizer theorem):

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

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

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

Corollary 1.8.15 (the orbit equation):

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

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

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

Corollary 1.8.16:

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

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

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

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

## The class equation

Definition 1.8.18:

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

${\mathcal {C}}_{G}(x):=G_{x}$ .

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

Lemma 1.8.19:

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

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

Corollary 1.8.20 (the class equation):

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

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

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

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

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

Lemma 1.8.22:

### p-groups

We shall now study the following thing:

Definition 1.8.24:

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

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

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

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

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

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

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

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