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/

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

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

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

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

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

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

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

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

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

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

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

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

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

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