no. 2
On minimal non-PC-groups
[Sur les non-PC-groupes minimaux]
Russo, Francesco1 ; Trabelsi, Nadir2
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.
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 
@article{AMBP_2009__16_2_277_0,
     author = {Russo, Francesco and Trabelsi, Nadir},
     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},
     year = {2009},
     doi = {10.5802/ambp.267},
     zbl = {1187.20042},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.267/}
}
TY  - JOUR
AU  - Russo, Francesco
AU  - Trabelsi, Nadir
TI  - On minimal non-PC-groups
JO  - Annales mathématiques Blaise Pascal
PY  - 2009
SP  - 277
EP  - 286
VL  - 16
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.267/
DO  - 10.5802/ambp.267
LA  - en
ID  - AMBP_2009__16_2_277_0
ER  - 
%0 Journal Article
%A Russo, Francesco
%A Trabelsi, Nadir
%T On minimal non-PC-groups
%J Annales mathématiques Blaise Pascal
%D 2009
%P 277-286
%V 16
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.267/
%R 10.5802/ambp.267
%G en
%F AMBP_2009__16_2_277_0
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/

[1] Beidleman, J. C.; Galoppo, A.; Manfredino, M. On PC-hypercentral and CC-hypercentral groups, Comm. Alg., Volume 26 (1998), pp. 3045-3055 | DOI | MR | Zbl

[2] Belyaev, V. V. Minimal non-FC-groups, VI All Union Symposium Group Theory (Čerkassy, 1978), Naukova Dumka, 1980, pp. 97-102 | MR | Zbl

[3] Belyaev, V. V.; Sesekin, N. F. Infinite groups of Miller-Moreno type, Acta Math. Hungar., Volume 26 (1975), pp. 369-376 | MR | Zbl

[4] Dilmi, A. Groups whose proper subgroups are locally finite-by-nilpotent, Ann. Math. Blaise Pascal, Volume 14 (2007), pp. 29-35 | DOI | Numdam | MR | Zbl

[5] Franciosi, S.; de Giovanni, F.; Tomkinson, M. J. Groups with polycyclic-by-finite conjugacy classes, Boll. U. M. I., Volume 7 (1990), pp. 35-55 | MR | Zbl

[6] Fuchs, L. Abelian Groups, Pergamon Press, London, 1967 | MR | Zbl

[7] Newman, M. F.; Wiegold, J. Arch. Math., Soviet Math. Dokl., Volume 15 (1964), pp. 241-250 | MR | Zbl

[8] Ol’shanskii, A. Yu. Infinite groups with cyclic subgroups, Soviet Math. Dokl., Volume 20 (1979), pp. 343-346 | MR | Zbl

[9] Otál, J.; Peña, J. M. Minimal Non-CC-Groups, Comm. Algebra, Volume 16 (1988), pp. 1231-1242 | DOI | MR | Zbl

[10] Polovickii, Ya. D. Groups with extremal classes of conjugated elements, Sibirski Math. Z., Volume 5 (1964), pp. 891-895 | MR

[11] Robinson, D. J. Finiteness conditions and generalized soluble groups, Springer Verlag, Berlin, 1972 | Zbl

[12] Tomkinson, M. J. FC-groups, Pitman, Boston, 1984 | MR | Zbl

[13] Trabelsi, N. On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups, C. R. Acad. Sci. Paris Ser. I, Volume 344 (2007), pp. 353-356 | MR | Zbl

[14] Xu, M. Groups whose proper subgroups are finite-by-nilpotent, Arch. Math., Volume 66 (1996), pp. 353-359 | DOI | MR | Zbl

Cité par Sources :