Let be a finite group which possesses an automorphism such that if and only if . If is the identity map on , prove that is abelian. Such an automorphism is called fixed point free of order 2. [Hint: Show that every element of can be written in the form and apply to such an expression.]
We define a mapping by .
Claim: is injective.
Proof of claim: Suppose . Then , so that , and . Then we have , hence . So is injective.
Since is finite and is injective, is also surjective. Then every is of the form for some .
Now let with . We have . Thus is in fact the inversion mapping, and we assumed that is a homomorphism. By a previous example, then, is abelian.