Let be a permutation group on the set (i.e. ), let , and let . Prove that . Deduce that if acts transitively on then .
First we show that .
Let . Then for some . Now ; hence .
Let . Then , so that . Hence .
Now if acts transitively on , the kernel of the action is by the previous exercise. Moreover, because , the homomorphism producing this action is injective, and thus has a trivial kernel. Thus .