I just recently started studying a little group theory, PH, so please, be gentle.

Since no one else has taken a stab. What the heck.

Assume G is a finite group and its order isn't divisible by 2, and

This is a homomorphism. If for , then the cycle group created by x is a subgroup of G of order 2.

Lagranges theorem says no way, since the order of G is not divisible by 2.

(I think.). And given y in G, there is one y in G where

Our hyp. says:

Make the appropriate cancellations:

We can write

Sorry, if I went off on a tangent or babble.