On minimal non-<i>PC</i>-groups

Russo, Francesco; Trabelsi, Nadir

doi:10.5802/ambp.267

On minimal non-PC-groups
[Sur les non-PC-groupes minimaux]

Russo, Francesco ¹ ; Trabelsi, Nadir ²

¹ Mathematics Department, University of Naples Federico II via Cinthia, Naples, 80126, Italy
² Laboratory of fundamental and numerical Mathematics, Mathematics Department University Ferhat Abbas, Setif, 19000, Algeria

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 277-286.

Résumé
Abstract

On dit qu’un groupe $G$ est un PC-groupe, si pour tout $x \in G$ , $G / C_{G} (x^{G})$ est une extension d’un groupe polycyclique par un groupe fini. Un non-PC-groupe minimal est un groupe qui n’est pas un PC-groupe mais dont tous les sous-groupes propres sont des PC-groupes. Notre principal résultat est qu’un non-PC-groupe minimal ayant un groupe quotient fini non-trivial est une extension cyclique finie d’un groupe abélien divisible de rang fini.

A group $G$ is said to be a PC-group, if $G / C_{G} (x^{G})$ is a polycyclic-by-finite group for all $x \in G$ . A minimal non-PC-group is a group which is not a PC-group but all of whose proper subgroups are PC-groups. Our main result is that a minimal non-PC-group having a non-trivial finite factor group is a finite cyclic extension of a divisible abelian group of finite rank.

Zbl

DOI : 10.5802/ambp.267

Classification : 20F24, 20F15, 20E34, 20E45
Mots-clés : Polycyclic-by-finite conjugacy classes, minimal non-PC-groups, locally graded groups.

Affiliations des auteurs :

Russo, Francesco ¹ ; Trabelsi, Nadir ²

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     title = {On minimal {non-\protect\emph{PC}-groups}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {277--286},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {2},
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Russo, Francesco; Trabelsi, Nadir. On minimal non-PC-groups. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 277-286. doi : 10.5802/ambp.267. http://www.numdam.org/articles/10.5802/ambp.267/

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