Publication date: 27 December 2023
Volume: Vol 58
Issue: Svezak 2
Pages: 317-326
DOI: 10.3336/gm.58.2.11
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.org/0000-0002-7030-4945
; Rochester Institute of Technology, Zagreb Campus, D.T. Gavrana 15, 10000 Zagreb, Croatia
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.
\(p\)-group, center, centralizer, Frattini subgroup, minimal nonabelian subgroup.
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