On totally semipermutable products of finite groups

Abstract

AbstractWe say a group G = AB is the totally semipermutable product of subgroups A and B if every Sylow subgroup P of A is totally permutable with every Sylow subgroup Q of B whenever $$ \gcd(|P|,|Q|)=1 $$ gcd ( | P | , | Q | ) = 1 . Products of pairwise totally semipermutable subgroups are studied in this article. Let $$ \mathfrak{U} $$ U denote the class of supersoluble groups and $$ \mathfrak{D} $$ D denote the formation of all groups which have an ordered Sylow tower of supersoluble type. We obtain the $$ \mathfrak{F} $$ F -residual of the product from the $$ \mathfrak{F} $$ F -residuals of the pairwise totally semipermutable subgroups when $$ \mathfrak{F} $$ F is a subgroup-closed saturated formation such that $$ \mathfrak{U}\subseteq \mathfrak{F}\subseteq \mathfrak{D} $$ U ⊆ F ⊆ D .