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An analogue of the Wielandt subgroup in infinite groups

  • Martyn R. DixonEmail author
  • Maria Ferrara
  • Marco Trombetti
Article
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Abstract

In this paper we define analogues of the Wielandt subgroup of a group. We say that a subgroup H of a group G is f-subnormal in G if there is a finite chain of subgroups \( H=H_0\le H_1\le \cdots \le H_n=G \) such that either \(|H_{i+1}: H_i|\) is finite or \(H_i\) is normal in \(H_{i+1}\), for \(0\le i\le n-1\). We study in a group G the connection between the subgroups \(\overline{\omega }(G)\) and \(\overline{\omega }_i(G)\) which are, respectively, the sets of elements of G normalizing all f-subnormal subgroups of G and those normalizing all infinite f-subnormal subgroups of G. In particular we show that \(\overline{\omega }_i(G)/\overline{\omega }(G)\) is always Dedekind and often abelian.

Keywords

f-subnormal subgroup Generalized f-Wielandt subgroup f-Wielandt subgroup Dedekind group Subsoluble group 

Mathematics Subject Classification

Primary: 20E15 Secondary 20F19 20F22 

Notes

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlabamaTuscaloosaUSA
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Napoli Federico II, Complesso Universitario Monte S. AngeloNaplesItaly

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