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Group actions and the class equation

Group Actions

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Definition: Let $G$ be a group and $X$ be a set. A (left) action of $G$ on $X$ is a map $G\times X\to X$ written $(g,x)\mapsto gx$ such that

  • $ex=x$ for all $x\in X$.
  • $g_1 (g_2) x = (g_1g_2)x$ for all $x\in X$ and $g_1, g_2\in G$.

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

Examples

  1. Let $G=S_{n}$ and \(X=\{1,2,\ldots,n\}\). A permutation $\sigma\in S_{n}$ is by definition a bijective map $\sigma: X\to X$. Then $\sigma i = \sigma(i)$ is an action of $G$ on $X$.

  2. Let $G$ be \(\mathrm{GL}_{2}(\mathbb{R})\) or the orthogonal group \(O(2)\) or the Euclidean group \(E(2)\). Then an element $g\in G$ acts by matrix multiplication on column vectors \(x\in\mathbb{R}^{2}\). More generally, \(\mathrm{GL}_{n}(\mathbb{R})\) and its subgroups act on \(\mathbb{R}^{n}\).

  3. Let $G$ be any group and let $X=G$ viewed as a set. Then $G$ acts on $X$ via left multiplication $(g,x)\mapsto gx$.

  4. Let $G$ be any group and let $X=G$ viewed as a set. Then $G$ acts on $X$ via conjugation $(g,x)\mapsto gxg^{-1}$.

  5. Let $G$ be any group and let $H$ be a subgroup. Then $G$ acts on the left cosets of $H$ by $(g,xH)\mapsto gxH$.

See this video and these notes

Definition: Let $X$ be a $G$-set. Define a relation $\sim$ on $X$ by $x\sim y$ if $x=gy$ for some $g\in G$. This relation is called $G$-equivalence. It is an equivalence relation and its equivalence classes are called orbits of $G$ on $X$.

Examples:

  1. Let $G=S_{n}$ and let $X={1,2,\ldots,n}$ with the “standard” action where $\sigma\cdot x = \sigma(x)$. Then every element of $X$ is $G$-equivalent to every other and there is only one orbit.

  2. Let $G=S_{n}$ and let $X=G$ with $G$ acting by conjugation. Thus two permutations $a$ and $b$ are $G$-equivalent if and only if they are conjugate by some $\sigma\in G$. We know that $a\sim b$ if and only if $a$ and $b$ have the same cycle decomposition, so the orbits are the possible cycle decompositions.

  3. Let $G$ be the dihedral group $D_{4}$ of symmetries of the square and let $X$ be the vertices of the square. Let $H$ be the subgroup of $G$ consisting of the identity and the reflection across one of the diagonals. Then there are three orbits. The two vertices off the diagonal are equivalent under $H$, but the two endpoints of the diagonal are fixed and are each an equivalence class on their own.

  4. Let $G$ be the orthogonal group $O(2)$ acting on $\mathbb{R}^{2}$. The orbits are the circles centered at the origin.

See this video and these notes

Lemma: Let $X$ be a $G$-set and let $x\in X$. The ssubset $G_x\subset G$ consisting of the elements $g\in G$ such that $gx=x$ is a subgroup called the “stabilizer subgroup of $x$”.

Theorem: Let $G$ be a finite group and $X$ a finite $G$-set. If $x\in X$, then the number of elements in the orbit of $x$ is the index $[G:G_x]$ of the stabilizer subgroup of $x$.

The class equation

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Lemma: Let $X$ be a finite $G$ set and let $X_{G}$ be the set of fixed points in $X$ (points where $gx=x$ for all $g\in G$.) Let $O_1,\ldots, O_k$ be the distinct orbits on $X$ – that is the partition of $X$ given by the equivalence relation $x\sim y$ when $y=gx$. Then \(|X| = |X_{G}|+\sum_{i=1}^{k} |O_{i}|.\)

Suppose now that $X=G$ with $G$ acting by conjugation. Then:

  • the fixed point set $X_G$ is the center of $G$.
  • The stabilizer of a point $g$ is the centralizer $C(g)$.
  • The orbit of a point $g$ is called the conjugacy class of $g$.

Let $x_1,\ldots, x_k$ be representatives for the conjugacy classes. Then \(|G| = |Z(G)| + \sum_{i=1}^{k} [G:C(x_i)]\)

This is called the class equation of $G$.

Corollary: The size of each conjugacy class is a divisor of the order of $G$.

Corollary: Suppose the order of $G$ is a power of a prime. Then the center of $G$ is nontrivial.

Corollary: Every group of order $p^2$ is abelian.

Burnside’s Theorem

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Theorem: Let $G$ be a finite group acting on a finite set $X$. For $g\in G$, let $X_g$ be the subset of $X$ consisting of elements fixed by $g$ (so that $gx=x$). Then \(k=\frac{1}{|G|}\sum_{g\in G} |X_{g}|\) where $k$ is the number of orbits of $G$ acting on $X$.

For applications to counting, see this video and this video as well as these notes.