University of IsfahanInternational Journal of Group Theory2251-76505320160901Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes6167877610.22108/ijgt.2016.8776ENMouniaBouchelaghemUniversity Setif 1NadirTrabelsiUniversity Setif 1Journal Article20141203A 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 $G$.https://ijgt.ui.ac.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf