Let f : G -> S(X) be a permutation representation. Fix x X. We can identify X with the set of cosets of Gx, provided f is transitive.

Theorem

Suppose that f is transitive. Then the map

t : G/Gx -> X,    t(gGx) = f(g)(x),

is a well-defined bijection and satisfies f(h) · t = t · Lh for every h G.

If, moreover, G is finite, then |G| = |Gx| · |X|.



The last assertion of the theorem says that, for finite groups G, the degree of a transitive permutation representation is equal to the index of a point stabilizer in G.