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?