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.

中文翻译：

---

关于具有半亚正态因子的群的超溶解性

摘要 如果存在子群 B 使得 G = A ⁢ B {G=AB} 且 AX 是 G 的子群，则群 G 的子群 A 在 G 中称为半正规。 我们引入新概念将次正规性和半正规性统一起来。如果 A 在 G 中低于正态或在 G 中为半正态，则 G 组的子组 A 被称为 G 中的半次正态。 A 组 G = A ⁢ B {G = AB} 研究了半次正态超可溶性子组 A 和 B。等式 G 𝔘 = ( G ′ ) 𝔑 {G^{\mathfrak{U}}=(G^{\prime})^{\mathfrak{N}}} 成立；此外，如果 G 中子群 A 和 B 的索引互质，则 G 𝔘 = G 𝔑 2 {G^{\mathfrak{U}}=G^{\mathfrak{N}^{2}}} 。这里 𝔑 {\mathfrak{N}} 、 𝔘 {\mathfrak{U}} 和 𝔑 2 {\mathfrak{N}^{2}} 分别是所有幂零、超溶和元幂等群的形成；