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Finite \(p\)-groups all of whose maximal subgroups either are metacyclic or have a derived subgroup of order \(\leq p\). (English) Zbl 1321.20017

A \(\mathcal P\)-group is a finite \(p\)-group whose maximal subgroups either are metacyclic or have a derived subgroup of order at most \(p\).
The authors classify, in turn: \(\mathcal P\)-groups with exactly one nonmetacyclic maximal subgroup; \(\mathcal P\)-groups with a metacyclic maximal subgroup and at least two nonmetacyclic maximal subgroups; and \(\mathcal P\)-groups all of whose maximal subgroups are nonmetacyclic. Thus a question of Y. Berkovich and Z. Janko [Groups of prime power order. Vol. 2. de Gruyter Expositions in Mathematics 47. Berlin: Walter de Gruyter (2008; Zbl 1168.20002)] is settled.
Amidst the above, the authors give a full classification of finite \(p\)-groups whose maximal subgroups all have derived subgroups of order at most \(p\).

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20E28 Maximal subgroups

Citations:

Zbl 1168.20002
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References:

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