The behaviour of a permutation representation f : G -> Sym(X) can be recorded inside G. The first step is to relate a point x of X to a particular subgroup Gx of G.

Definition

If x X, then the stabilizer of x is the subgroup Gx of G given by

Gx = {g G | g(x) = x}.

If g(x) = x, then g is said to fix or stabilize x.

The second step is to construct permutation representations from within G. Suppose H is a subgroup of G. We shall construct a transitive permutation representation of G on G/H with H as point stabilizer.

Theorem

For g element of G, let Lg : G/H -> G/H be given by

Lg(hH) = ghH.

Then

L : G -> Sym(G/H),    g -> Lg

is a transitive permutation representation. Moreover, the stabilizer of the element H of G/H is H.


We are now ready for the final step. It will establish that any transitive permutation representation G -> Sym(X) can be identified with the permutation representation L as above for H the stabilizer of an element x of X.