The Sylow Subgroup Exchange And Subgroups Weak S-can Make Up The Structure Of A Finite Group Influence

This paper has two aspects:1. We determine the structure of finite group whose Sylow subgroups are all two-generator and abelian. We obtain the following theorem:Let G be a finite group. If Sylow subgroups of G are all two-generator group and abelian, then G satisfies one of the following:(i) Let D be a minimal non-abelian group whose order is pq2, if G is an odd order group and D-free, then G is a supersolvable group;(ii) If G is a non-solvable group, then G has a section L2(9);(iii) If G is A4-free, p is a minimal prime, divisor G, then G is p-nilpotent;(iv) If G is a solvable group, then G is M-group and D2-group.2. We study the relationship between the structure of finite groups and weakly s-supple-mented properties of some subgroups. We obtain some sufficient conditions on p-nilpotency of finite groups by using the weakly s-supplemented properties of some minimal subgroups, cyclic subgroups of order 4, subgroups of order p2.