@article{Hu_Huang_Adarchenko_2021, title={On $\Pi$-permutable subgroups in finite groups}, volume={73}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/768}, DOI={10.37863/umzh.v73i10.768}, abstractNote={<p>UDC 512.542</p> <p>Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $\Pi$ be a nonempty subset of the set $\sigma.$<span class="Apple-converted-space">&nbsp;&nbsp;</span>A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{complete Hall $\Pi$-set} of $G$ if every member of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $\sigma _{i}\in \Pi$ and ${\cal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma _{i}\in \Pi$ such that $\sigma_i\cap\pi(G) eq\varnothing.$<span class="Apple-converted-space">&nbsp;</span>A subgroup $A$ of $G$ is called<span class="Apple-converted-space">&nbsp;</span>(i) {${\cal H}^{G}$-<em>permutable</em>} if $AH^{x}=H^{x}A$ for $H\in {\cal H}$ and $x\in G$;<span class="Apple-converted-space">&nbsp;</span>(ii) {$\Pi$-<em>permutable</em> in $G$} if $A$ is ${\cal H}^{G}$-permutable for some complete Hall $\Pi$-set $\cal H$ of $G.$<span class="Apple-converted-space">&nbsp;</span></p> <p>We study the influence of $\Pi$-permutable subgroups on the structure of $G.$<span class="Apple-converted-space">&nbsp;</span>In particular, we prove that if $\pi= \displaystyle\bigcup olimits_{\sigma_{i}\in \Pi} \sigma_{i}$ and $G =AB,$ where $A$ and $B$ are ${\cal H}^{G}$-permutable $\pi$-separable (respectively, $\pi$-closed) subgroups of $G,$ then $G$ is also $\pi$-separable (respectively, $\pi$-closed).&nbsp;<span class="Apple-converted-space">&nbsp;</span>Some known results are generalized.</p&gt;}, number={10}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Hu, B. and Huang, J. and Adarchenko, N. M.}, year={2021}, month={Oct.}, pages={1423-1431} }