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I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra special $p$ groups.

We can always choose a central subgroup $N$ of order $p$ in a finite nilpotent group as well. Is something similar possible for finite solvable groups?

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No to both questions.

For example for the first, in the generalized quaternion groups, there is a unique element of order $2$, which is thus contained in any non-trivial subgroup.

For the second, consider either of the groups the group $S_3$ or $S_4$ which are solvable but have trivial centers.

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  • $\begingroup$ But i hope the class of $p$ groups($p$ odd) for which we can choose a cyclic central subgroup which is not contained in the commutator is large. $\endgroup$ – user114539 Jun 30 '15 at 19:18
  • $\begingroup$ @user114539 What would "large" mean? Please ask a more precise question then. $\endgroup$ – Tobias Kildetoft Jun 30 '15 at 19:19

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