Let G be a group acting on a set S containing at least two elements Assume that G is transitive that is, given any x,y S, there exists g G such that gx y Prove (a) for x S, the orbit x of x is S (b) all the stabilizers G x , (for x S) are conjugate (c) if G has the property g G gx x for all x S (e ...

a For any x in S by the transitivity of the action there exists g in G such that gx y for any y in S ...

Let G be a group acting on a set S containing at least two elements. Assume that

Question:

Let G be a group acting on a set S containing at least two elements. Assume that G is transitive; that is, given any x,y ϵ S, there exists g ϵ G such that gx = y. Prove
(a) for x ϵ S, the orbit x̄ of x is S;
(b) all the stabilizers G_x, (for x ϵ S) are conjugate;
(c) if G has the property: {g ϵ G| gx = x for all x ϵ S} = (e (which is the case if G < S_n for some n and S = {1,2, ... , n}) and if N ⊲ G and N < G_x' for some x ϵ S, then N = (e);
(d) for x ϵ S,|S| = [G: G_x]; hence |S| divides |G|.