On the supersolubility of a group with semisubnormal factors

Abstract

Abstract A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that G = A ⁢ B {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 = A ⁢ B {G=AB} with semisubnormal supersoluble subgroups A and B is studied. The equality G 𝔘 = ( G ′ ) 𝔑 {G^{\mathfrak{U}}=(G^{\prime})^{\mathfrak{N}}} is established; moreover, if the indices of subgroups A and B in G are relatively prime, then G 𝔘 = G 𝔑 2 {G^{\mathfrak{U}}=G^{\mathfrak{N}^{2}}} . Here 𝔑 {\mathfrak{N}} , 𝔘 {\mathfrak{U}} and 𝔑 2 {\mathfrak{N}^{2}} are the formations of all nilpotent, supersoluble and metanilpotent groups, respectively; H 𝔛 {H^{\mathfrak{X}}} is the 𝔛 {\mathfrak{X}} -residual of H. Also we prove the supersolubility of G = A ⁢ B {G=AB} when all Sylow subgroups of A and of B are semisubnormal in G.