Glasnik Matematicki, Vol. 58, No. 2 (2023), 317-326. $$

$CZ$-GROUPS WITH NONABELIAN NORMAL SUBGROUP OF ORDER $p^4$

Mario Osvin Pavčević and Kristijan Tabak

Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia
e-mail:mario.pavcevic@fer.hr

Rochester Institute of Technology, Zagreb Campus, D.T. Gavrana 15, 10000 Zagreb, Croatia
e-mail:kxtcad@rit.edu

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 $\mathrm{\Phi }(G).$ 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.

2020 Mathematics Subject Classification.   20D15, 20D25

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

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

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, 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, 2010.
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4. M. Hall, Jr., Theory of groups, The Macmillan Company, New York, 1959.
   MathSciNet

5. M. O. Pavčević, and K. Tabak, CZ-groups, Glas. Mat. Ser. III 51(71) (2016), 345–358.
   MathSciNet    CrossRef