Group Theory/Cardinality identities for finite representations

Definition (permutation representation):

Let $G$ be a group and let $X$ be a set. A permutation representation of $G$ on $S$ is a representation $\rho :G\to \operatorname {Aut} (S)$, where the automorphisms of $S$ are taken in the category of sets (that is, they are just bijections from $S$ to itself).

Definition (pointwise stabilizer):

Let $G$ be a group, let ${\mathcal {A}}$ be an algebraic variety and let $A$ be an instance of ${\mathcal {A}}$. Suppose that $\rho :G\to \operatorname {Aut} (A)$ is a representation in the category defined by ${\mathcal {A}}$. Let $S\subseteq A$. Then the pointwise stabilizer of $S$ is given by

$G_{S}:=\{g\in G|\forall a\in S:ga=a\}$.

Proposition (transitive permutation representation is equivalent to right multiplication on quotient by stabilizer):

Let $G$ be a group, let $X$ be a set and suppose that we have a permutation representation $\pi :G\to \operatorname {Sym} (X)$ which is transitive. Let $x\in X$ be arbitrary and let $G_{x}\leq G$ be the pointwise stabilizer of $x$. Consider the action $\rho :G\to \operatorname {Sym} (G/G_{x})$ by left multiplication, where $G/G_{x}$ is the set of left cosets of $G_{x}$ (which is in fact never a normal subgroup in this situation, unless the action is trivial, because $G_{gx}=g^{-1}G_{x}g$). Then there exists a $G$-isomorphism from $X$ to $G/G_{x}$.

Proof: We define $f:G/G_{x}\to X$ as follows: $gG_{x}$ shall be mapped to $gx$. First, we show that this map is well-defined. Indeed, suppose that we take $h\in G_{x}$. Then $ghG_{x}$ is mapped to $ghx=x$. Then we note that the map is surjective by transitivity. Finally, it is also injective, because whenever $gx=hx$, we have $h^{-1}gx=x$ by applying $h^{-1}$ to both sides and using a property of a group action, and thus $h^{-1}g\in G_{x}$, that is to say $gG_{x}=hG_{x}$. That $f(ghG_{x})=g(hx)$ follows immediately from the definition, so that we do have an isomorphism of representations. $\Box$

We are now in a position to derive some standard formulae for permutation representations.

Theorem (orbit-stabilizer theorem):

Let $G$ be a group, and let $\pi :G\to \operatorname {Sym} (X)$ be a permutation representation on a set $X$. Then

$|Gx|=[G:G_{x}]$.

Proof: $G$ acts transitively on $Gx$. The above $G$-isomorphism between $Gx$ and $G/G_{x}$ is bijective as an isomorphism in the category of sets. But the notation $[G:G_{x}]$ stood for $|G/G_{x}|$. $\Box$

Theorem (class equation):

Let $G$ be a finite group and let $\pi :G\to \operatorname {Sym} (X)$ be a permutation representation on the finite set $X$. Then

$|X|=\sum _{j=1}^{n}[G:G_{x_{j}}]$,

where $Gx_{1},\ldots ,Gx_{n}$ are the orbits of $G$, and $Gx_{k}\neq Gx_{j}$ for $k\neq j$. (We also say that $x_{1},\ldots ,x_{n}$ are a system of representatives for the orbits of $G$.)

Proof: $G$ acts transitively on each orbits, and the orbits partition $X$. Hence, by the orbit-stabilizer theorem,

$|X|=\sum _{j=1}^{n}|Gx_{j}|=\sum _{j=1}^{n}[G:G_{x_{j}}]$. $\Box$

Definition (fixed point set):

Let $G$ be a group that acts on a set $X$, and let $S\subseteq G$ be a subset of $G$. Then the fixed point set of $S$ is defined to be

$\operatorname {Fix} (S):=\{x\in X|\forall s\in S:sx=x$.