# Proving Non-Simplicity

## Homework Statement

Let G be a finite group. Prove that if some conjugacy class has exactly two elements then G can't be simple

## The Attempt at a Solution

I originally proved this accidentally assuming that G is abelian, which it isn't. So say x and y are conjugate then we have conjugacy classes {e} and {x,y}. I know that the size of a conjugacy class has to divide the order of a group so |G| is even and must be greater than 2. I'm having trouble showing the existence of a normal subgroup though.