Some Questions About ρ-Nilpotency And Supersolvability Of Finite Groups

The main results of this paper mainly contains two sections, which centre on the following two important parts of finite group theory respectively:1. To study the relations between the weak supplement of subgroups and nilpotency or supersolvability of finite groups.2. To discuss the relations between the maximal or minimal subgroups of sylow subgroups and supersolvable groups.In the first section, namely §3.1, we introduce the following weak supplement of a subgroup for the first time:Definition: let G be a define group, if is a subgroup of the G(H ≤ G). While we definite H has a weak supplement in G, if there exists a proper subgroup K of G such that G = HK.By this concept, we analyze p-nilpotency and supersolvability of finite groups.Apart from this, under the condition of s-semipermutation, seminormality and C-normality, we analyze the relations between the maximal or minimal subgroups of Sylow subgroups and supersolvable groups in the second section, namely §3.2.Based on these, we improve and extend the relevant results to saturated formation including the class of all supersoluble groups. Consequently, more new results are obtained.