On factorized groups with permutable subgroups of factors

TL;DR

This paper studies groups that are factorized by two subgroups with a special permutability property, proving that their product is supersoluble if the subgroups are supersoluble, advancing understanding of group factorizations.

Contribution

It introduces and investigates the concept of msp-permutable subgroups and proves the supersolubility of their product in certain conditions.

Findings

The product of two supersoluble msp-permutable subgroups is supersoluble.

msp-permutability generalizes mutual permutability for Sylow subgroups.

The paper characterizes groups with factorization by msp-permutable subgroups.

Abstract

The subgroups $A$ and $B$ of a group~$G$ are called {\rm msp}-permutable, if the following statements hold: $AB$~is a subgroup of~$G$; the subgroups $P$ and $Q$ are mutually permutable, where $P$~is an arbitrary Sylow $p$-subgroup of~$A$ and $Q$~is an arbitrary Sylow $q$-subgroup of~$B$, ${p\neq q}$. In the present paper, we investigate groups that factorized by two {\rm msp}-permutable subgroups. In particular, the supersolubility of the product of two supersoluble {\rm msp}-permutable subgroups is proved.