On the supersolubility of a group with semisubnormal factors

Abstract

A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semisubnormal in G if A is subnormal in G or seminormal in G. A group G = AB with semisubnormal supersoluble subgroups A and B is studied. The equality Gᵁ = (G΄)ᶰ is established; moreover, if the indices of subgroups A and B in G are relatively prime, then Gᵁ = (G΄)ᶰ². Here N, U and N² are the formations of all nilpotent, supersoluble and metanilpotent groups, respectively; H ͯ is the X-residual of H. Also we prove the supersolubility of G = AB when all Sylow subgroups of A and of B are semisubnormal in G.