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16.13 (2006)

Solved

Does there exist a finite $p$-group $G$ all of whose maximal subgroups $H$ are special, that is, satisfy $Z(H) = [H, H] = \Phi(H)$?

Contributor: Ya. G. Berkovich

Progress

Yes, for all primes there are groups of arbitrarily large size with this property (J. Cossey, Bull. Austral. Math. Soc., 89 (2014), 415–419); an example for $p = 2$ given by a Sylow 2-subgroup of $L_3(4)$ was independently presented by V. I. Zenkov at Mal'cev Meeting–2014, 10–14 November, 2014, Novosibirsk.

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