Glasnik Matematicki, Vol. 51, No. 2 (2016), 345-358.

CZ-GROUPS

Kristijan Tabak and Mario Osvin Pavčević

Department of applied mathematics, Faculty of Electrical Engineering and Computing , University of Zagreb , 10 000 Zagreb, Croatia
e-mail: mario.pavcevic@fer.hr

Abstract.   We describe some aspects of the structure of nonabelian p-groups G for which every nonabelian subgroup has a trivial centralizer in G, i.e. only it's center. We call such groups CZ-groups. The problem of describing the structure of all CZ-groups was posted as one of the first research problems in the open problems list in Yakov Berkovich's book 'Groups of prime power order' Vol 1 ([1]). Among other features of such groups, we prove that a minimal CZ-group must contain at least p⁵ elements. The structure of maximal abelian subgroups of these groups is described as well.

2010 Mathematics Subject Classification.   20D15, 20D25.

Key words and phrases.   p-group, center, centralizer, Frattini subgroup, minimal nonabelian subgroup.

DOI: 10.3336/gm.51.2.05

References:

1. Y. Berkovich Groups of Prime Power Order Vol. 1., Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
   MathSciNet    

2. Y. Berkovich and Z. Janko, Groups of Prime Power Order Vol. 2., Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
   MathSciNet    

3. Y. Berkovich and Z. Janko, Groups of Prime Power Order Vol. 3., Walter de Gruyter GmbH & Co. KG, Berlin, 2011.
   MathSciNet