Let be a group and a nonempty set. Let act on . Prove that if and for some , then . Deduce that if acts transitively on , then the kernel of the action is .
First we prove that .
If , then . Now , so that . Hence .
Let . Then , so that . Hence .
Suppose now that the action of on is transitive; that is, for all , there exists such that . We show that, if is the kernel of the action, .
Let . Then for all . Let . Then , so that for all . Thus for all , hence .
Let , and let . Because the action of on is transitive, we have for some . Now for some , thus . Hence stabilizes ; since is arbitrary, .