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 ${\displaystyle }$.

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

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 ${\displaystyle }$.

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.