Actions

Overview

Let $X$ be a set and $G$ be a group. A left action of $G$ on $X$ is a map $\ast :G\times X\to X$ such that

1. Identity. $ex=x$ for all $x\in X$,
2. Compatibility. $(g_1{g}_2)x=g_1(g_{2}x)$ for all $x\in X$ and all $g_1,g_2\in G$.

Under these conditions, $X$ is called a left $G$-set. Analogous definitions exist for right actions and right $G$-sets. In both cases, we say $G$ acts on $X$.

Application of currying is used to convert the foregoing definition into an equivalent one using a homomorphism. In particular, let $G$ be a group and $X$ be a set. Then we define a group action of $G$ on $X$ as any homomorphism $\varphi :G\to \text{Sym}(X)$.

We can relate these two definitions by introducing identity $\varphi (g)(x)=gx$.

We say $G$ acts faithfully on $X$ if $\text{Ker}(\varphi )=\{e\}$. That is, the identity element in $G$ is the only element that leaves each $x\in X$ fixed.

Sub-G-Sets

Let $X$ be a left $G$-set. Then a left sub-$G$-set is a set $Y\subseteq X$ such that $Gy\subseteq Y$ for all $y\in Y$. An analogous definition exists for a right sub-$G$-set.

Equivalently, $Y$ must equal the union of some subset of the orbits of $X$ under $G$.

Induced Actions

Let $H$ be a subgroup of $G$. Commonly induced actions include:

• $G$ is a left $H$-set under action of $h\in H$ on $g\in G$ given by $hg$.
  • $G$ is transitive if and only if $H=G$.
• $G/H$ is a left $G$-set under action of $g\in G$ on $xH\in G/H$ given by $g(xH)=(gx)H$.
  • $G/H$ is always transitive.

Isomorphisms

Let $X$ and $Y$ be $G$-sets. An isomorphism between $X$ and $Y$ is a sets/functions#Bijectivity map $\varphi :X\to Y$ that satisfies the following property for all $g\in G$ and $x\in X$: $$g\phi(x) = \phi(gx).$$

Stabilizers

Let $X$ be a $G$-set and let $x\in X$. The stabilizer (subgroup) of $x$, defined as $G_x=\{g\in G\mid gx=x\}$, is a subgroup of $G$. It is also known as the isotropy subgroup of $x$.

We similarly define $X_g=\{x\in X\mid gx=x\}$.

Orbits

Let $X$ be a $G$-set. Each cell in the partition of the equivalence relation $\sim$ given as $$\forall a, b \in X, a \sim b \iff \exists g \in G, ga = b$$

is an orbit in $X$ under $G$. If $x\in X$, the cell containing $x$ is the orbit of $x$, denoted $Gx$. In other words, $$Gx = { gx \mid g \in G }.$$

Transitivity

A group $G$ is transitive on $G$-set $X$ if for each $x_1,x_2\in X$, there exists $g\in G$ such that $g{x}_1=x_2$. In other words, if there is exactly one orbit of $X$ under $G$.

If $\varphi :G\to \text{Sym}(X)$ is our associated group action, $G$ is transitive on $X$ if and only if $\varphi [[G]]$ is transitive on $X$.

Orbit-Stabilizer Theorem

Let $X$ be a $G$-set and let $x\in X$. Then $Gx$ is isomorphic to $G/G_x$. Sometimes the theorem is more simply stated as a measure of cardinalities: $|Gx|=(G:G_x)$. If $G$ is finite, Lagrange's theorem then shows $|Gx|=|G|/|G_x|$.

Every transitive $G$-set $X$ is isomorphic to a left coset $G$-set $G/H$ where $H\le G$.

Every $G$-set is equal to the union of its orbits. Since every orbit is a transitive sub-$G$-set, every $G$-set is isomorphic to a disjoint union of left coset $G$-sets (which itself is also a $G$-set).

Furthermore, $G/H\simeq G/K$ as $G$-sets if and only if $H$ and $K$ are conjugate subgroups of $G$.