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December 3rd, 2017, 07:50 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, 04:04 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, 02:48 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 02:54 AM.
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December 15th, 2017, 11:33 AM   #4
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Found an Easier Way

Actually, you do not need the Isomorphism THM machinery to do this. We have that N is a normal subgroup of G. So if a is in N, every conjugate of a by an element of N is in N.

Hence, constructing a conjugacy class equation for N makes sense. Since N is bigger than (1), you now do the same proof as that which shows that a nontrivial p-group has a nontrivial center. (That is the case when N = G.)
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December 16th, 2017, 08:15 AM   #5
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Well done. If you want to seriously study groups, factor groups should become second nature for you.
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December 21st, 2017, 05:00 PM   #6
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They are. Abstract Algebra was my favorite topic in grad school, which was decades ago. I now am getting into math history and the interesting (and sometimes crazy) people who got involved.
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