Let $G$ be a finite group such that $G'\cap Z(G)\neq 1$. Suppose also that $G'$ is an elementary abelian $p$-group; $G'\nleq Z(G) $; $(G/Z(G))'$ is a minimal normal subgroup of $G/Z(G)$.
Can we deduce that $(G/Z(G))'\cap Z(G/Z(G))\neq 1$?
A question about intersection of center and commutator subgroup
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abstract-algebra
group-theory
finite-groups
examples-counterexamples
p-groups
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1Neat question though,! I would find it very interesting to hear an answer to "When does $G'\cap Z(G))\not= 1$ imply that $(G/Z(G))'\cap Z(G/Z(G))'\not= 1$?" – 2012-12-21
1 Answers
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No, we can't. Minimal counterexample: $G=\text{SmallGroup}(96,197)$.
In here, $G'\cong \mathbb{Z}_2\times\mathbb{Z}_2\times \mathbb{Z}_2$ and $Z(G)\cong \mathbb{Z}_2$.
$G/Z(G)\cong\text{SmallGroup}(48,49)$, and $(G/Z(G))'\cong \mathbb{Z}_2\times\mathbb{Z}_2$ is a minimal normal subgroup of $G/Z(G)$. We have that $Z(G/Z(G))\cong \mathbb{Z}_2\times\mathbb{Z}_2$ as well, but the two intersect trivially.
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0First of all, Thanks for your answer. Here's my real problem [1](http://v3rgil.altervista.org/Video/1.jpg) [2](http://v3rgil.altervista.org/Video/2.jpg) [3](http://v3rgil.altervista.org/Video/3.jpg) [4](http://v3rgil.altervista.org/Video/4.jpg) [5](http://v3rgil.altervista.org/Video/5.jpg) [6](http://v3rgil.altervista.org/Video/6.jpg) [7](http://v3rgil.altervista.org/Video/7.jpg) . My question refers to the last picture. I obviously miss a condition that runs everything... Finding it would provide an answer also to your question above (at least in some particular case). – 2012-12-22