Glasnik Matematicki, Vol. 47, No. 2 (2012), 325-332.

FINITE p-GROUPS ALL OF WHOSE MAXIMAL SUBGROUPS, EXCEPT ONE, HAVE ITS DERIVED SUBGROUP OF ORDER ≤ p

Zvonimir Janko

Abstract.   Let G be a finite p-group which has exactly one maximal subgroup H such that |H'|>p. Then we have d(G)=2, p=2, H' is a four-group, G' is abelian of order 8 and type (4,2), G is of class 3 and the structure of G is completely determined. This solves the problem Nr. 1800 stated by Y. Berkovich in [3].

2010 Mathematics Subject Classification.   20D15.

Key words and phrases.   Finite p-groups, minimal nonabelian p-groups, commutator subgroups, nilpotence class of p-groups, Frattini subgroups, generators and relations.

DOI: 10.3336/gm.47.2.08

References:

1. Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin-New York, 2008.
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2. Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 2, Walter de Gruyter, Berlin-New York, 2008.
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3. Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 3, Walter de Gruyter, Berlin-New York, 2011.
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