On some groups whose subnormal subgroups are contranormal-free

Document Type : Ischia Group Theory 2022

Authors

1 Department of Algebra and Geometry, School of Mathematics and Mechanics, National Dnipro University, Gagarin Prospect 72, Dnipro 10, 49010 Ukraine

2 Department of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy

Abstract

If $G$ is a group, a subgroup $H$ of $G$ is said to be contranormal in $G$ if $H^G = G$, where $H^G$ is the normal closure of $H$ in $G$. We say that a group is contranormal-free if it does not contain proper contranormal subgroups. Obviously, a nilpotent group is contranormal-free. Conversely, if $G$ is a finite contranormal-free group, then $G$ is nilpotent. We study (infinite) groups whose subnormal subgroups are contranormal-free. We prove that if $G$ is a group which contains a normal nilpotent subgroup $A$ such that $G/A$ is a periodic Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is generated by subnormal nilpotent subgroups; in particular $G$ is a Baer group. Furthermore, if $G$ is a group which contains a normal nilpotent subgroup $A$ such that the $0$-rank of $A$ is finite, the set $\Pi(A)$ is finite, $G/A$ is a Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is a Baer group.

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References

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International Journal of Group Theory
Volume 14, Issue 2 - Serial Number 2
Proceedings of the Ischia Group Theory 2022-Part 3
June 2025
Pages 99-115

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History

  • Receive Date: 15 September 2023
  • Revise Date: 24 April 2024
  • Accept Date: 27 April 2024
  • Published Online: 05 May 2024

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