On $Π$-permutable subgroups in finite groups - Hu, B.; Huang, J.; Adarchenko, N. M. (Ukrains’kyi Matematychnyi Zhurnal, 2026)
UDC 512.542Let $σ =\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $Π$ be a nonempty subset of the set $σ.$ A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a complete Hall $Π$-set of $G$ if every member of ${\cal H}$ is a Hall $σ _{i}$-subgroup of $G$ for some $σ _{i}\in Π$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $σ _{i}\in Π$ such that $σ_i\capπ(G) eq\varnothing.$ A subgroup $A$ of $G$ is called (i) {${\cal H}^{G}$-permutable} if $AH^{x}=H^{x}A$ for $H\in {\cal H}$ and $x\in G$; (ii) {$Π$-permutable in $G$} if $A$ is ${\cal H}^{G}$-permutable for some complete Hall $Π$-set $\cal H$ of $G.$ We study the influence of $Π$-permutable subgroups on the structure of $G.$ In particular, we prove that if $π= \displaystyle\bigcup olimits_{σ_{i}\in Π} σ_{i}$ and $G =AB,$ where $A$ and $B$ are ${\cal H}^{G}$-permutable $π$-separable (respectively, $π$-closed) subgroups of $G,$ then $G$ is also $π$-separable (respectively, $π$-closed). Some known results are generalized.