Suppose that G is a finite, nonabelian group with odd order. Show s is surjective, and hence bijective.
I have been told to look at the effects of the squaring map, $s\colon G\to G$, defined by $s(g)=g^2$ on the elements of cyclic groups $\langle g\rangle$ of $G$.
I’m stumped. Could anyone give me a nudge in the right direction or (being hopeful) a full solution?
Thanks a lot.
On each cyclic subgroup $C$ of $G$, the $s$ map is a homomorphism. Prove that $s$ is injective by considering $\ker s$ in $C$. It is here that you use that $G$ of odd order.