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December 3rd, 2017, 08:50 AM  #1 
Newbie Joined: Mar 2014 Posts: 16 Thanks: 2  Normal Subgroups of pgroups
Let G be a pgroup and N ≠ (1) be a normal subgroup of G. Then N and Z(G) have a nontrivial intersection. Is this an induction problem?

December 3rd, 2017, 05:04 PM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
Since you mention induction, I assume that G is finite; aside: I realized I don't know if this is true for infinite pgroups. 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 pgroup of order less than the order of G the statement is true. Now the center of any finite pgroup is nontrivial; 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 nontrivial. 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. 
December 4th, 2017, 03:48 AM  #3 
Newbie Joined: Mar 2014 Posts: 16 Thanks: 2 
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. 
December 15th, 2017, 12:33 PM  #4 
Newbie Joined: Mar 2014 Posts: 16 Thanks: 2  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 pgroup has a nontrivial center. (That is the case when N = G.) 
December 16th, 2017, 09:15 AM  #5 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
Well done. If you want to seriously study groups, factor groups should become second nature for you.

December 21st, 2017, 06:00 PM  #6 
Newbie Joined: Mar 2014 Posts: 16 Thanks: 2 
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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