How many elements of order 7 are there in a simple group of order 168?
By Sylow's 3rd theorem the number of Sylow's 7-subgroups of a group of order 168 is either 1 or 8. It cannot be 1 because it would imply such a Sylow 7-subgroup must be normal contradicting the fact the group is simple. Therefore there are 8 Sylow 7-subgroups. Given to be two Sylow-7 subgroups with then it implies (because is a subgroup of and my Lagrange's theorem divides - a prime). This means the non-trivial elements in are all distinct. And furthermore each one generates the group i.e. have order 7. There are a total of of such elements. And conversely any element of order seven gives rise to a Sylow -subgroup and must be included among those elements. Therefore there are 48 elements of order 7.