Let G be group acting on the nonempty set I Let H lessthanor

Let G be group acting on the non-empty set I. Let H lessthanorequalto G and i elementof I. Suppose that H acts transitively on I. (a) Let g elementof G. Show that there exists h elementof H with gi = hi. (b) Put K:= Stab_G(i). Show that G = HK.

Solution

Definition: A group H acts on a set A (we use A instead of I, for clarity) if given any x, y in A there exists a h in H such that h(x) =y.

(1) Let G,H,g,i be as in the question.

Take y =g(i) and x =i as above.

As the action of H is transitive , it follows that there exists h in H such that h(i) =g(i) , as required.

(ii) From (1) there exists h such that g(i) =h(i)

which implies gh^-1 (i) =i

By defintion, gh^-1 belongs to K =Stab(i)

So gh^-1=k

Thus g=hk , h in H and k in K.

This shows G =HK , as required