# Show the intersection of a nonidentity normal subgroup and the center of P is not trivial

P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial.

By the class equation, I proved that Z(P)is not 1. Then, how do prove I the intersection of M and the center of P is not 1 or empty?

Thank you very much for your time...

• What do you mean by “nontrivial union”? – k.stm Dec 13 '14 at 11:42
• I meant that it is not not 1 or empty.@k.stm – Calico Dec 13 '14 at 11:46
• I think you are confusing "$\cap =$ intersection" with "$\cup =$ union" – Myself Dec 13 '14 at 11:47
• Yep, I meant intersection. My bad. @Myself. – Calico Dec 13 '14 at 11:48
• In you proof that Z(P) > 1 you have probably considered the action of P on itself by conjugation. This time, you have a normal subgroup M. So you could try to consider the action of P on M by conjugation and try to repeat the argument with some modifications. – Myself Dec 13 '14 at 11:53

$M$, being normal, is the union of conjugacy classes (with respect to $P$), meaning a conjugacy class lies completely in $M$ or is disjoint from $M$. Since the size of a conjugacy class is either $1$ or a multiple of $p$ the number of singleton classes in $M$ is a multiple of $p$ ($M$ is also a p-group), moreover it is not $0$ since the class of the identity is one of them. So there are at least $p$ singleton classes in M. But these singletons are central elements in P, which proves the assertion. With thanks to @Myself for his comment.