1. The problem statement, all variables and given/known data Let G be a group with order [tex]\left| G \right| = 60[/tex]. Assume that G is simple. Now let H be the set of all elements that can be written as a product of elements of order 5 in G. Show that H is a normal subgroup of G. Then conclude that H = G 2. Relevant equations 3. The attempt at a solution I started by proving that H acutally is a subgroup. I've then shown that there are 6 Sylow-5 subgroups in G and that they are cyclic. I know that all the elements of order 5 are the generators of the Sylow-5 subgroups. But how I can use that to show that H is normal escapes me. All help/hints appreciated :).