1. The problem statement, all variables and given/known data If some conjugacy class of an element in a group G contains precisely two elements, show that G cannot be a simple group. 3. The attempt at a solution This question was longer, with two questions before this one which I could answer and which probably lead to the answer on this question. I showed that for an element x in G the elements of G which commute with x, form a subgroup C(x) of G, called the centralizer. I also proved that the size of the conjugacy class of x is equal to the number of left cosets of C(x) in G. But now for the last question, I need a hint.