On the Supersoluble Residual of a Product of Supersoluble Subgroups

Abstract

Let ℙ be the set of all primes. A subgroup H of a group G is called ℙ-subnormal in G, if either H = G, or there exists a chain of subgroups H = H₀ ≤ H₁ ≤ . . . ≤ Hn = G, with |Hi : Hi-1| ∈ ℙ for all i. A group G = AB with ℙ-subnormal supersoluble subgroups A and B is studied. The structure of its supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the derived subgroup of G. Besides, if the indices of the subgroups A and B are coprime, then the supersoluble residual coincides with the intersection of the metanilpotent residual of G and all normal subgroups of G such that all corresponding quotients are primary or biprimary. From here new signs of supersolubility are derived.