Is it possible to find a non-abelian group (preferably a finite $p$-group) with the following property?
If M is a maximal subgroup of a group (or finite $p$-group) $G$, then $Z(M)\not \le Z(G)$.
2026-03-25 10:55:28.1774436128
Center of a maximal subgroup
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Yes, it is possible. There are two non-abelian groups of order $p^3$ (up to isomorphism). Due to Sylow theorem they both have a subgroup of order $p^2$ (it is well-known, that all groups of order $p^2$ are abelian) and it is not very difficult to prove that their centres have order $p$.