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December 3rd, 2017, 08:50 AM   #1
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Normal Subgroups of p-groups

Let G be a p-group and N ≠ (1) be a normal subgroup of G. Then N and Z(G) have a nontrivial intersection. Is this an induction problem?
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December 3rd, 2017, 05:04 PM   #2
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Since you mention induction, I assume that G is finite; aside: I realized I don't know if this is true for infinite p-groups. Here's most of a solution that uses a minimal counterexample, a form of induction:

Assume false and let G be a minimal counterexample; i.e. for any p-group of order less than the order of G the statement is true. Now the center of any finite p-group is non-trivial; easy proof via the class equation. So let $Z=Z(G)$ and $Z_2/Z=Z(G/Z)$. By assumption $(NZ\cap Z_2)/Z=(N\cap Z_2)Z/Z$ is not trivial; i.e. $N\cap Z_2$ is non-trivial. Let $1\neq z\in N\cap Z_2$. Then for any $g\in G$, $[g,z]=g^{-1}z^{-1}gzZ=Z$ in $G/Z$. That is $[g,z]\in Z\cap N=<1>$ or $1\neq z\in N\cap Z$, contradiction.
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December 4th, 2017, 03:48 AM   #3
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It is assumed to be a finite group - left that out. I thought the Isomorphism THM would be used - but couldn't get the key move.

Last edited by dpsmith; December 4th, 2017 at 03:54 AM.
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