On dit qu’un groupe est un PC-groupe, si pour tout , 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 is said to be a PC-group, if is a polycyclic-by-finite group for all . 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.
Mots clés : Polycyclic-by-finite conjugacy classes, minimal non-PC-groups, locally graded groups.
@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 -
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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