Question

Let G be a finite group.let S={g€G|g^5=e}, where e is identity element of G.show that |S| is odd.

Answer 1

Since $g^{-1}=g$ only for $g=e$ , all $g\in S$ are divided into pairs except $g=e$ . Then the size of S is 2n+1, where n is number of pairs.