Sufficient and necessary conditions for supersolvable group;

Using the weakly c-normality of minimal subgroup of to characterize the structure of a finite group,some sufficient conditions of finite supersolvable group are obtained.

In this paper we prove:Let F be a saturated formation containing U,the class of all supersolvable groups and suppose that G is a group with a normal subgroup H such that G/H∈F.

The completely conditional permutability of minimal subgroups is used in this paper to obtain the sufficient condition for supersoluble groups: let G be a finite group, if every minimal subgroup and every cyclic subgroup of order 4 of G is completely c-permutable in G, G is supersoluble.

Using the concept,X-permutable subgroup of finite groups,two sufficient conditions of supersoluble groups are obtained:(1) G is a soluble group,let X be a subgroup of G such that X contains every maximal subgroup and every minimal subgroup of G.

In this paper,by using the concept of permutable subgroups,a new criterion of supersolubility of the product G=AB of finite supersoluble groups A and B is given.

This paper assumes that every non-cyclic Sylow subgroup P of G has a subgroup D such that 1<|D|<|P| and all subgroups H of P with order |H|=|D| and with 2|D|(if P is a non-abelian 2-group and |P:D|>2) are normally embedded in G,and some sufficient conditions are obtained on G to be p-nilpotent groups and supersolvable groups.

This paper gives all the nilpotent subgroups and solvable subgroups of symmetric group S5.