# 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 $G$)
* $\forall x\in X:f(\iota ,x)=x$ and
* $\forall \sigma ,\tau \in G,x\in X:f(\sigma ,f(\tau ,x))=f(\sigma \tau ,x)$.
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
* $\forall x\in X:\iota x=x$ and
* $\forall \sigma ,\tau \in G,x\in X:\sigma (\tau x)=(\sigma \tau )x$.
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:
* $S_{n}$ acts on $\mathbb {R} ^{n}$ via $\sigma (x_{1},\ldots ,x_{n})=(x_{\sigma (1)},\ldots ,x_{\sigma (n)})$.
* $GL_{n}(\mathbb {R} )$ 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
* faithful iff $(\forall x\in X:\sigma x=\tau x)\Rightarrow \sigma =\tau$ ('identity on all elements of $x$ enforces identity on $G$')
* 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
* 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.