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Definition: A Schmidt group is defined to be a non-nilpotent finite group with the property that every proper subgroup is nilpotent. Also, a group is said to be a $\mathscr{B}$-group if any independent generating sequence has the same length.

According to McDougall-Bagnall and Quick's result, $G$ is a $\mathscr{B}$-group iff one the following holds:

  1. $G$ is a p-group for some prime $p$.
  2. $G = P \rtimes Q$ where P is a $p$-group and $Q$ is a cyclic $q$-group ( $p$, $q$ are distinct primes) such that $C_Q (P) \neq Q$ and every non-trivial element of $Q/C_Q(P)$ acts fixed-point-freely on $P$/$\Phi(P)$ (where $\Phi(P)$ denotes the Frattini subgroup of $P$).

One known fact about a Schmidt group is that it is a semi-direct product of a normal Sylow $p$-subgroup and a non-normal cyclic Sylow $q$-subgroup. This poses a natural question whether Schmidt groups satisfy $2$., hence implies being $\mathscr{B}$-groups.

I truly appreciate any experience, guide or suggestion for this question.

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