Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 4 2 2023 07 01 On some groups without perfect factors 79 94 172327 10.30504/jims.2023.400933.1123 EN L. A. Kurdachenko Department of Algebra and Geometry , School of Mathematics and Mechanics, Dnipro P. Longobardi Department of Mathematics ,University of Salerno, Fisciano (SA) Italy M. Maj Department of Mathematics, University of Salerno, Fisciano (SA), Italy Journal Article 2023 06 08 If $G$ is a group and $H, K$ are normal subgroups of $G$, $H\leq K$, then $K/H$ is said to be a $G$-perfect factor if $[K/H, G] = K/H$. If $G$ is a nilpotent group, then every non-trivial factor of $G$ is not $G$-perfect. Conversely, if $G$ is finite and all non-trivial factors of $G$ are not $G$-perfect, then $G$ is nilpotent. We study (infinite) groups with no non-trivial $G$-perfect factors. We prove that if either $G$ is a locally generalized radical group with finite section rank, or $G$ has a normal nilpotent subgroup $A$ such that $G/A$ is a locally finite group with Chernikov Sylow $p$-subgroups for every prime $p$, and $G$ has no non-trivial $G$-perfect factors, then for every prime $p$ there exists a positive integer $s_p$ such that $\zeta_{s_p}(G)$, the $s_p$-term of the upper central series of $G$, contains the Sylow $p$-subgroups of $G$, and $G/Tor(G)$ is nilpotent. In particular, $G$ is hypercentral and the hypercentral length of $G$ is at most $\omega+k$, for some positive integer $k$. https://jims.ims.ir/article_172327_b5bfb6cd2f5cc1d06569c45d595393ea.pdf