Group Actions

Group Action

Let $G$ be a group and $X$ a set. An action of $G$ on $X$ is a function $G\times X\to X,(g,x)\mapsto g\cdot x$ satisfying

• $1\cdot x=x$ for all $x\in X$, and
• $(gh)\cdot x=g\cdot (h\cdot x)$ for all $x\in X$ and $g,h\in G$

A set $X$ equipped with an action of $G$ is called a $G$-set.

e.g.

• $G$ (the group) acts on $G$ (the set) by left multiplication.
• Let $H\le G$. Then $G$ acts on $X=G/H=\{gH\mid g\in G\}$ by left multiplication.
• $G$ acts on itself by conjunction.

Proposition

A group action of $G$ on a set $X$ is the same as homomorphism $G\to \mathrm{Sym}(X)$.

Proof Given an action $G\times X\to X$ defineed, for every $g\in G$, a map ${\rho }_g:X\to X$ by ${\rho }_g(x)=g\cdot x$. Then ${\rho }_g$ is a bijection because ${\rho }_{g^{-1}}$ is its inverse. Therefore, we get a map $\rho :G\to \mathrm{Sym}(X),g\mapsto {\rho }_g$Now check that $\rho$ is a homomorphism. Indeed, $\rho (g)\rho (h)={\rho }_g\circ {\rho }_h={\rho }_{gh}=\rho (gh)$Conversely, given such $\rho$, define $G\times X\to X$ by $(g,x)\mapsto {\rho }_g(x)$. This is a group action and the two constructions we defined are inverses to each other. $\square$

Orbit and Stabiliser

Let $G$ be a group acting on a set $X$ and let $x\in X$. Then, the orbit of $x$ is the subset $\mathcal{O}(x)=\{g\cdot x\mid g\in G\}\subset X$. The stabilizer of $x$ in $G$ is the subgroup $G_x={\mathrm{Stab}}_G(x)=\{g\mid g\cdot x=x\}\le G$.

Classification of Actions

Transitive Action

A group action is transitive if it has only one orbit. Equivalently, this means for any two elements $x,y\in X$, there exists an element $g\in G$ such that $g\cdot x=y$.

Free Action

A group action of $G$ on $X$ is said to be free if $g\cdot x=x$ implies $g=1$ for all $x\in X$.

Faithful (Effective) Action

An action of $G$ on $X$ is said to be faithful if the only element of $G$ that acts as the identity on $X$ is the identity element of $G$.

Proposition

An action of a group $G$ on a set $X$ defines an equivalence relation on $X$: $x\sim y⟺\exists g\in G,y=g\cdot x$

The Orbit-Stabilizer Theorem

Conjugacy Classes

When $G$ acts on itself by conjugation, the orbits are called conjugacy classes.

Centraliser

The centralizer of an element $x$ in a group $G$, is the set $C_G(x)$ of elements that commute with $x$. Or equivalently, the stabilizer of $x\in G$ under conjugation is the centralizer $C_G(x)$ of $x$.

Orbit-Stabilizer Theorem

Let $G$ be a group acting on a set $X$ . Let $x\in X$ and let $H=G_x$ be the stabilizer of $x$ in $G$ and let $\mathcal{O}(x$) be the orbit. Then the the map $\phi :G/H\to \mathcal{O}(x),gH\mapsto gx$is a bijection. In particular, if $G$ is finite then $|G|=|G_x||\mathcal{O}(x)|$. So the size of every orbit divides the order of the group.

Proof The map is surjective: an inverse image of $gx\in \mathcal{O}(x)$ is $gH$. The map is injective since: $gx=hx⟺h^{-1}gx=x⟺h^{-1}g\in G_x=H⟺gH=hH$

The Class Equation

Let $G$ be a finite group. Then $|G|={\sum }_i[G:C_G(x_i)].$

Proof $G$ acts on itself by conjugation. It follows that $G$ is a disjoint union of conjugacy classes. For every class $\mathcal{O}(x)$ we have $|\mathcal{O}(x)|=[G:C_G(x)]$ as by Orbit-Stabilizer Theorem we have $\mathcal{O}(x)\cong G/C_G(x)$.

$p$-group

A finite group is called a $p$-group if its order is a $p$-power.

Corollary

Every $p$-group has a non-trivial centre.

Proof Let $G$ be a finite $p$−group and make it act on itself by conjugation. Observe that:$\begin{array}{r}|\mathcal{O}(x)|=1⟺x\in Z(G)\end{array}$By class equation, we know that $|\mathcal{O}(x)|=[G:C_G(x)]$.

Lemma

For any group $G$, $G/Z(G)$ is cyclic iff $G$ is abelian.

Cauchy’s Theorem

If $p$ is a prime dividing the order of a finite group $G$ then $G$ contains an element of order $p$.

Proof