Author:
Pavčević Mario Osvin, ,Tabak Kristijan,
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.
Publisher
University of Zagreb, Faculty of Science, Department of Mathematics
Reference5 articles.
1. Y. Berkovich, Groups of prime power order. Vol. 1, Walter de Gruyter, Berlin-New York, 2008.
2. Y. Berkovich, Z. Janko, Groups of prime power order. Vol. 2, Walter de Gruyter, Berlin-New York, 2008.
3. Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 3, Walter de Gruyter, Berlin-New York, 2010.
4. M. Hall, Jr., Theory of groups, The Macmillan Company, New York, 1959.
5. M. O. Pavčević, and K. Tabak, CZ-groups, Glas. Mat. Ser. III 51(71) (2016), 345-358.