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A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  28 March 2018

M. DE FALCO ,
F. DE GIOVANNI ,
C. MUSELLA  and
N. TRABELSI
M. DE FALCO
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli, Italy email mdefalco@unina.it
F. DE GIOVANNI*
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli, Italy email degiovan@unina.it
C. MUSELLA
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli, Italy email cmusella@unina.it
N. TRABELSI
Affiliation:
Department of Mathematics, University of Setif 1, Setif 19000, Algeria email nadir_trabelsi@yahoo.fr
*
✉degiovan@unina.it
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Abstract

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If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.

Keywords

FC-groupnilpotent groupEngel group

MSC classification

Primary: 20F24: FC-groups and their generalizations
Secondary: 20F45: Engel conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 1 , August 2018 , pp. 24 - 33
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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