Minimal CSS-supplemented subgroups of finite groups

Abstract

Over the years, group theorists have evaluated the impact of studying supplemented subgroups on the theory of finite groups. As a result, they have described dozens of criteria for nilpotency and solubility that depend on various supplemented embedded classes of subgroups. In keeping with this approach, we have also established a new concept of CSS-supplemented subgroups. A subgroup H of a group G is described as a CSS-supplemented subgroup in G if T ⊵ G such that G = HT and H ∩ T is SS-supplemented in G. A subgroup H of G is known as an SS-supplemented subgroup in G if there is a subgroup T of G such that HT = G and H ∩ T is SS-quasinormal in T. The SS-supplemented subgroup is expanded based on this agreement by the CSS-supplemented subgroup. By describing new criteria involving formation, we have examined the minimal CSS-subgroup of G based on the structure of the SS-supplemented subgroup that affected p-solubility and p-nilpotency.