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$R$-conjugate-permutable subgroup; nilpotent group; quasinilpotent group; Sylow subgroup
A subgroup $H$ of a finite group $G$ is said to be conjugate-permutable if $HH^{g}=H^{g}H$ for all $g\in G$. More generaly, if we limit the element $g$ to a subgroup $R$ of $G$, then we say that the subgroup $H$ is $R$-conjugate-permutable. By means of the $R$-conjugate-permutable subgroups, we investigate the relationship between the nilpotence of $G$ and the $R$-conjugate-permutability of the Sylow subgroups of $A$ and $B$ under the condition that $G=AB$, where $A$ and $B$ are subgroups of $G$. Some results known in the literature are improved and generalized in the paper.
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