Proof: Let , then clearly . But, since is cyclic we have that
and for some . Thus, we have that for some (this is because cosets form a partition). And so
(since ). The conclusion follows.
Now since we have, by Lagrange's theorem that . If it's the first or the last we're done. So, assume that it could be either or . Then
and since all prime ordered groups are cyclic it follows from the lemma that is abelian and so which contradicts the assumption that .
The conclusion follows.