On the Supersoluble Residual of a Product of Supersoluble Subgroups

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 = H_0 \leq H_1 \leq... \leq H_n = G, with |H_i: H_{i-1}| \in 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.

Abstract licence: CC BY-NC

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