Definition – Group Action – Math-reference.net

Contents

1. Introduction
2. Definitions
3. Orbits, sending set, fixed set and stabilisers.
4. Conjugation

Definition – Group action

Let $\langle G,*\rangle$ be a group, and let $X$ be a set. Also, let $f:X\times{G} \rightarrow X$ be a function under the rule $f(g,x) = g*x$ for $g\in G$ . We say that $G$ acts on the set $X$ if the following conditions hold,

$f(e_G, x) = x\text{,} \quad \forall x\in X$ .
The operation $*$ is associative, ie $f(g_1*g_2,x) = f(g_1,f(g_2,x))\text{,}\quad \forall g_1,g_2\in G$ .

It can be proven that a group action is simply a rearrangement of the elements of $X$ .

Orbits, sending set, fixed set and stabilisers.

Given a group action (as defined above), we can define the following sets,

$\text{orb} (x) = \{f(g,x) = y\in X~|~g\in G \}$ – Ie, all the elements that $x$ is mapped to under the group action. We refer to this set as the orbit of $x$ .
$\text{stab}(x) = \{g\in G ~|~ f(g,x)=x\}$ . We refer to this set as the stabiliser of $x$ .
$\text{fix}(g) = \{x\in X ~|~ f(g,x) = x\}$ . We refer to this set as the fixed set of $g$ .
$\text{send}_x(y) = \{g\in G ~|~ f(g,x) = y\}$ . We refer to the set as the sending set of $x$ to $y$ .

There is

See the orbit stabilizer theorem + orbit counting theorem + some others you need to add.

Conjugacy Classes

We may define a special group action which deserves its own definition. Let $\langle G,* \rangle$ be a group, and let $X =G$ . Let $g\in G$ and $h\in G$ . We refer to elements of the form $h*g*h^{-1}$ as conjugates (respectively “a conjugate”) of the element $g\in G$ .

Let us define a group action $f$ under the rule

$f(g,x) = g*x*g^{-1}$

Ie, $x$ is mapped to the conjugate corresponding to $g\in G$ . We may use this group action to define an equivalence relation on the set $G$ , where for any $a,b \in G$ ,

$aRb \iff \exists g\in G ~ \text{s.t} ~ a = f(g,b) = g*b*g^{-1}$

We can prove that this relation partitions the set $G$ using the equivalence relation conditions. See here for the details.

This is of course a formal proof, but in fact, it should be clear as to why the conjugacy classes partition $G$ . The answer in rearranging the condition for conjugation as a group action,

$f(g,x)*g = g*x$

Let us set $y = f(g, x) \in G$ . Then, we have

$y*g = g*x$

But, the left and right cosets, $yG = gG$ both partition the set $G$ , and so it follows that all elements of $G$ must be contained in both cosets. Let $y*g = g*x = t \in G$ be such an element. Then, this proves that $t$ can be written in the form $y*g = g*x = t$ , and so via the group axioms we have that any element $y$ may be written as

$y=g*x*g^{-1}$

But, as every element can be written in this form, it makes sense that the elements can be “categorised” based on being written in this form, or at least to wonder if that is the case, which is the motivation for proving so using the equivalence relation conditions.

$\mathbf{conj_G(x)}$ and $\mathbf{cent_G(x)}$

For the group action as defined in conjugacy classes, we have alternative names for the orbit and stabiliser of the group action. We define,

$conj_G(x) = orb(x) = \{f(x,g) ~|~ g\in G \}$

And,

$cent_G(x) = stab(x) = \{g\in G ~|~ f(g,x)=x \}$

Where $conj_G(x)$ is called the conjugacy class of $x$ , and $cent_G(x)$ is referred to as the centraliser of $x$ in $G$ .