On partially conjugate-permutable subgroups of finite groups

Let $R$ be a subset of a group $G$. We call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup of $G$, if $HH^{x}=H^{x}H$ for all $x\in R$. This concept is a generalization of conjugate-permutable subgroups introduced by T. Foguel. Our work focuses on the influence of $R$-conjugate-permutable subgroups on the structure of finite groups in case when $R$ is the Fitting subgroup or its generalizations $F^{*}(G)$ (introduced by H. Bender in 1970) and $\tilde{F}(G)$ (introduced by P. Shmid 1972). We obtain a new criteria for nilpotency and supersolubility of finite groups which generalize some well known results.