Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes

Document Type : Research Paper

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University Setif 1

Abstract

A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $x\in G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.

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References

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International Journal of Group Theory
Volume 5, Issue 3 - Serial Number 3
September 2016
Pages 61-67

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History

  • Receive Date: 03 December 2014
  • Revise Date: 18 March 2015
  • Accept Date: 18 March 2015
  • Published Online: 01 September 2016

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