TY - JOUR
ID - 8776
TI - Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes
JO - International Journal of Group Theory
JA - IJGT
LA - en
SN - 2251-7650
AU - Bouchelaghem, Mounia
AU - Trabelsi, Nadir
AD - University Setif 1
Y1 - 2016
PY - 2016
VL - 5
IS - 3
SP - 61
EP - 67
KW - Polycyclic-by-finite conjugacy classes
KW - minimal non-(PF)C-group
KW - minimal non-FC-group
KW - PrÃ¼fer rank
DO - 10.22108/ijgt.2016.8776
N2 - A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $x\in G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.
UR - https://ijgt.ui.ac.ir/article_8776.html
L1 - https://ijgt.ui.ac.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf
ER -