ok, since the class equation is:
and since Z(G) ≠ G, and ,
we have either:
15 = 1 + 3k + 5m
15 = 3 + 3k + 5m
15 = 5 + 3k + 5m
we want to show that only the first one is possible.
suppose 15 = 3 + 3k + 5m.
clearly m = 0,1 or 2.
now, this means that 15 - 5m is divisible by 3, so m = 0. this means that for all non-central g. so contains no elements of order 3, which is impossible since all of Z(G) is in .
on the other hand, suppose that:
15 = 5 + 3k + 5m.
then k = 0,1,2 or 3, so 15 - 3k is divisible by 5, so k = 0. then for every non-central g, which again leads to a contradiction.