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Glasnik matematički, Vol. 58 No. 2, 2023.

Original scientific paper

https://doi.org/10.3336/gm.58.2.11

\(CZ\)-groups with nonabelian normal subgroup of order \(p^4\)

Mario Osvin Pavčević ; Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia
Kristijan Tabak orcid id orcid.org/0000-0002-7030-4945 ; Rochester Institute of Technology, Zagreb Campus, D.T. Gavrana 15, 10000 Zagreb, Croatia

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Abstract

A \(p\)-group \(G\) with the property that its every nonabelian subgroup has a trivial centralizer (namely only its center) is called a \(CZ\)-group.
In Berkovich's monograph (see [1]) the description of the structure of a \(CZ\)-group was posted as a research problem. Here we provide further progress on this topic based on results proved in [5]. In this paper we have described the structure of \(CZ\)-groups \(G\) that possess a nonabelian normal subgroup of order \(p^4\) which is contained in the Frattini subgroup \(\Phi(G).\) We manage to prove that such a group of order \(p^4\) is unique and that the order of the entire group \(G\) is less than or equal to \(p^7\), \(p\) being a prime. Additionally, all such groups \(G\) are shown to be of a class less than maximal.

Keywords

\(p\)-group, center, centralizer, Frattini subgroup, minimal nonabelian subgroup.

Hrčak ID:

312018

URI

https://hrcak.srce.hr/312018

Publication date:

22.12.2024.

Visits: 342 *

Publication date: 27 December 2023

Volume: Vol 58

Issue: Svezak 2

Pages: 317-326

DOI: 10.3336/gm.58.2.11

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: \(p\)-group, center, centralizer, Frattini subgroup, minimal nonabelian subgroup.

\(CZ\)-groups with nonabelian normal subgroup of order \(p^4\)

Mario Osvin Pavčević[1]

Email: mario.pavcevic@fer.hr

Kristijan Tabak[2]

Email: kxtcad@rit.edu

 

[1] Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia

[2] Rochester Institute of Technology, Zagreb Campus, D.T. Gavrana 15, 10000 Zagreb, Croatia

Abstract

A \(p\)-group \(G\) with the property that its every nonabelian subgroup has a trivial centralizer (namely only its center) is called a \(CZ\)-group. In Berkovich's monograph (see [1]) the description of the structure of a \(CZ\)-group was posted as a research problem. Here we provide further progress on this topic based on results proved in [5]. In this paper we have described the structure of \(CZ\)-groups \(G\) that possess a nonabelian normal subgroup of order \(p^4\) which is contained in the Frattini subgroup \(\Phi(G).\) We manage to prove that such a group of order \(p^4\) is unique and that the order of the entire group \(G\) is less than or equal to \(p^7\), \(p\) being a prime. Additionally, all such groups \(G\) are shown to be of a class less than maximal.

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