Annales Mathématiques
Blaise Pascal
no. 2
On minimal non-PC-groups
[Sur les non-PC-groupes minimaux]
Francesco Russo1 ; Nadir Trabelsi2
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.

On dit qu’un groupe G est un PC-groupe, si pour tout x∈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∈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.

DOI : 10.5802/ambp.267
Classification : 20F24, 20F15, 20E34, 20E45
Mots clés : Polycyclic-by-finite conjugacy classes, minimal non-PC-groups, locally graded groups.
Francesco Russo 1 ; Nadir Trabelsi 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 
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Francesco Russo; Nadir Trabelsi. On minimal non-PC-groups. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 277-286. doi : 10.5802/ambp.267. https://ambp.centre-mersenne.org/articles/10.5802/ambp.267/

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