On totally semipermutable products of finite groups

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