On minimal non-PC-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

Annales mathématiques Blaise Pascal (2009)

  • Volume: 16, Issue: 2, page 277-286
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Russo, Francesco, and Trabelsi, Nadir. "On minimal non-PC-groups." Annales mathématiques Blaise Pascal 16.2 (2009): 277-286. <http://eudml.org/doc/10580>.

@article{Russo2009,
abstract = {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.},
affiliation = {Mathematics Department, University of Naples Federico II via Cinthia, Naples, 80126, Italy; Laboratory of fundamental and numerical Mathematics, Mathematics Department University Ferhat Abbas, Setif, 19000, Algeria},
author = {Russo, Francesco, Trabelsi, Nadir},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Polycyclic-by-finite conjugacy classes; minimal non-PC-groups; locally graded groups; PC-groups; minimal non-PC groups; polycyclic-by-finite groups; subgroups of finite index},
language = {eng},
month = {7},
number = {2},
pages = {277-286},
publisher = {Annales mathématiques Blaise Pascal},
title = {On minimal non-PC-groups},
url = {http://eudml.org/doc/10580},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Russo, Francesco
AU - Trabelsi, Nadir
TI - On minimal non-PC-groups
JO - Annales mathématiques Blaise Pascal
DA - 2009/7//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 2
SP - 277
EP - 286
AB - 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.
LA - eng
KW - Polycyclic-by-finite conjugacy classes; minimal non-PC-groups; locally graded groups; PC-groups; minimal non-PC groups; polycyclic-by-finite groups; subgroups of finite index
UR - http://eudml.org/doc/10580
ER -

References

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  9. J. Otál, J. M. Peña, Minimal Non-CC-Groups, Comm. Algebra 16 (1988), 1231-1242 Zbl0644.20025MR939041
  10. Ya. D. Polovickii, Groups with extremal classes of conjugated elements, Sibirski Math. Z. 5 (1964), 891-895 MR168658
  11. D. J. Robinson, Finiteness conditions and generalized soluble groups, (1972), Springer Verlag, Berlin Zbl0243.20033
  12. M. J. Tomkinson, FC-groups, (1984), Pitman, Boston Zbl0547.20031MR742777
  13. N. Trabelsi, On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups, C. R. Acad. Sci. Paris Ser. I 344 (2007), 353-356 Zbl1113.20032MR2310669
  14. M. Xu, Groups whose proper subgroups are finite-by-nilpotent, Arch. Math. 66 (1996), 353-359 Zbl0857.20015MR1383898

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