1. The problem statement, all variables and given/known data Prove in any finite group G, the number of elements not equal to their own inverse is an even number. 2. Relevant equations if ab = ba = e, then a = b-1 and b = a-1 3. The attempt at a solution Let S, A, B, be subsets of G where S = A + B. Let a ∈ A s.t. there exists a unique b ∈ B so that ab = ba = e and a =/= b. Then |A| = |B| = k, k ∈ ℤ. Then |S| = |A| + |B| = k + k = 2k. And the definition of even is 2k, so |S| will always be even.  How can I make this better?