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
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 $$G \to S_X$$.

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

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: $$A x := A x$$, where the first juxtaposition stands for the group action definition and the second for matrix multiplication.

Types of actions
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 $$x \in X$$.

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

We now attempt to characterise these three definitions; i.e. we try to find conditions equivalent to each.

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

Proof:

A group acts on itself faithfully via left multiplication. Hence, by the previous theorem, there is a monomorphism $$G \to S_G$$.

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

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

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 $$\tau^{-1} \sigma \in G_y = \{\iota\}$$.

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

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 $$y \in G(x)$$.

Regarding the stabilizers we have the following two theorems:

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 $$\sigma^{-1} \in G_x$$.

Proof:


 * $$\begin{align}

\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{align}$$

Cardinality formulas
The following theorem will imply formulas for the cardinalities of $$G_x$$, $$|G|$$, $$(G:G_x)$$ or $$X$$ respectively.

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 $$G_x \le G$$, $$\tau^{-1} \sigma \in G_x$$. 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.

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 $$(G:G_x) = \frac{|G|}{|G_x|}$$.

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.

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

The following lemma, which is commonly known as Burnside's lemma, is actually due to Cauchy:

The class equation
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: $$x \in Z(G) \Leftrightarrow \forall \sigma \in G : \sigma x \sigma^{-1} = x \Leftrightarrow G(x) = \{x\}$$.

Proof: This follows from lemma 1.8.19 and Corollary 1.8.16.

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.

Lemma 1.8.22:

p-groups
We shall now study the following thing:

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) $$\Phi\left(g_1,\Phi(g_2,v)\right)=\Phi(g_1g_2,v)$$ 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.