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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...

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  • $\begingroup$ What do you mean by “nontrivial union”? $\endgroup$ – k.stm Dec 13 '14 at 11:42
  • $\begingroup$ I meant that it is not not 1 or empty.@k.stm $\endgroup$ – Calico Dec 13 '14 at 11:46
  • $\begingroup$ I think you are confusing "$\cap = $ intersection" with "$\cup =$ union" $\endgroup$ – Myself Dec 13 '14 at 11:47
  • $\begingroup$ Yep, I meant intersection. My bad. @Myself. $\endgroup$ – Calico Dec 13 '14 at 11:48
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    $\begingroup$ 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. $\endgroup$ – Myself Dec 13 '14 at 11:53
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$M$, 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.

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