# Group Theory/Cardinality identities for finite representations

Definition (permutation representation):

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

Definition (pointwise stabilizer):

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

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

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

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

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

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

Theorem (orbit-stabilizer theorem):

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

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

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

Theorem (class equation):

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

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

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

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

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

Definition (fixed point set):

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

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