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.