TITLE: Finite $G$ with involutory automorphism $\alpha$ with no nontrivial fixed points. Proving properties of $\alpha$.
QUESTION [6 upvotes]: The question at hand is:

Let G be a finite group and $\alpha$ an involutory automorphism of G, which doesn't fixate any element aside from the trivial one. 
  1) Prove that $ g \mapsto g^{-1}g^{\alpha} $ is an injection 
  2) Prove that $\alpha$ maps every element to its inverse 
  3) Prove that G is abelian 

I think I've found 1).
I assume $ g_1^{-1}g_1^{\alpha} = g_2^{-1}g_2^{\alpha} $ and from this I get $ (g_2g_1^{-1})^\alpha = g_2g_1^{-1} $, so $g_2 = g_1$ (is this correct?) 
For 2) I'm sort of stumped though, not sure how to start proving that, so any help please?

REPLY [3 votes]: 1) is correct as you've written. 3) is an easy consqeuence of 2), so I'll leave that part to you.
For 2) we need to use the fact that $\alpha$ is involutory... Note that $\varphi:g\mapsto g^{-1}g^\alpha$ is injective, so it is bijective (because $G$ is finite). Consider an arbitrary $g$, and $h:=\varphi^{-1}(g)$. What is $\varphi^2(h)$ (in terms of $h$ first, then in terms of $g$)?