Group Actions

Definition.
Let $G$ be a group and $X$ be a set. We say that $G$ acts on $X$ if there is a map $\cdot: G\times X\to X$ such that:

This action induces a group homomorphism

\[\rho: G\to S_X\]

where $\rho(g): X\to X$ is the map given by left multiplication by $g$.

Definition.
Suppose a group $G$ acts on a set $X$. For $x\in X$ and $g\in G$, we define:

Definition.
Let $G$ act on $X$. We say:

The action is transitive if there is only one orbit.
The action is faithful if $g\cdot x = x$ for all $x$ implies $g=e$.
Equivalently, $\rho: G\to S_X$ is injective.

In general,

\[\ker \rho = \bigcap_{x\in X} G_x,\]

and $G/\ker \rho$ acts faithfully on $X$.

Theorem (Orbit–Stabilizer Theorem).
Let $G$ act on $X$, and let $x\in X$. Then $G_x$ is a subgroup of $G$, and there is a bijection

\[G/G_x\ \longrightarrow\ G\cdot x,\qquad gG_x \mapsto g\cdot x.\]

Theorem (Burnside’s Lemma).
Let $G$ be a finite group acting on a finite set $X$. Then

\[|G\backslash X| = \frac{1}{|G|}\sum_{g\in G} |X^g|.\]

Proof (Sketch).
Let $S = \{(g,x)\in G\times X : g\cdot x = x\}$. Count $|S|$ in two ways.

Theorem (Class Equation).
Let $G$ act on $X$. Then

\[|X| = |X^G| + \sum_{\substack{G\cdot x\in G\backslash X \\ |G\cdot x|>1}} [G : G_x].\]

Example 1.
Let $G$ act on itself by left multiplication:

The action is transitive (one orbit).
The stabilizer of any element is trivial.
The action is faithful, giving an embedding $\rho: G\hookrightarrow S_G$ (Cayley’s Theorem).

Example 2.
Let $G$ act on itself by conjugation:

The orbit of $x\in G$ is its conjugacy class:
\[\mathrm{Cl}(x) = \{gxg^{-1}: g\in G\}.\]
The stabilizer of $x$ is the centralizer:
\[C_G(x) = \{g\in G: gx = xg\}.\]
The fixed points of $g$ are exactly $C_G(g)$.
The fixed points of $G$ form the center:
\[Z(G) = \{g\in G: gx = xg \ \forall\, x\in G\}.\]
Class Equation:
\[|G| = |Z(G)| + \sum [G : C_G(g)],\]
where the sum runs over representatives of the conjugacy classes not contained in $Z(G)$.

Example 3.
Let $G$ act on the left coset space $G/H$ by left multiplication:

The action is transitive (one orbit).
The stabilizer of $xH$ is $xHx^{-1}$.
The kernel of $\rho: G\to S_{G/H}$ is the normal core:
\[\mathrm{Cor}(H) = \bigcap_{x\in G} xHx^{-1},\]
the largest subgroup of $H$ that is normal in $G$.

Example 4.
Let $X = \{H \leq G\}$ be the set of all subgroups of $G$. Let $G$ act on $X$ by conjugation:

The stabilizer of $H$ is the normalizer:
\[N_G(H) = \{g\in G : gHg^{-1} = H\},\]
which is the largest subgroup of $G$ in which $H$ is normal.
The orbit of $H$ is the set of conjugates $\{gHg^{-1} : g\in G\}$. By the Orbit–Stabilizer Theorem, the number of conjugates of $H$ is $[G : N_G(H)]$.

Easy:

Let $G$ be a finite $p$-group. Show that $Z(G) \neq \{e\}$. (Hint: Use the class equation.)
(UCI 2023 Jan Algebra Qual) Let $G$ be a group of order $p^2$. Show that $G$ is abelian. Classify all such groups up to isomorphism.
Let $G$ be a finite group, and $H\leq G$ with $[G : H] = p$ where $p$ is the smallest prime divisor of $|G|$. Show that $H \trianglelefteq G$. (Hint: Apply Example 3 and show that $\mathrm{Cor}(H) = H$.)
(UCI 2024 June Algebra Qual) Prove: If $H$ has finite index $n$ in $G$, then there exists a normal subgroup $K \trianglelefteq G$ with $K \leq H$ and $[G : K] \leq n!$. Do not assume $G$ is finite.

Medium:

Let $G$ be a group acting transitively on a finite set $X$ with $|X| > 1$. Show that there exists $g\in G$ with no fixed points (i.e., $X^g = \varnothing$).
Let $G$ be a finite group, and $H\leq G$ be a proper subgroup. Show that $G\neq \bigcup_{g\in G} gHg^{-1}$. (Hint: Consider $G$ acting on $G/H$ via left translation. There is some $g_0$ without any fixed point. Show that $g_0\notin \bigcup_{g\in G}gHg^{-1}$)