The θ-Pairs For Maximal Subgroups Of Finite Groups

In the paper, we make a study of the solvability ,supersolvability of the group,with the aid of O-pairs of less maximal subguoups.Especially we define some sets of special maximal subgroups . For example,Fs(G) notes the set that does not contain solvable residue; a(G) is the set that does not contain supersolvable residue, and normal indexes of maximal subgroups have not square divisors , and so on. In addition , we make combination with conditions of O- pairs for maximal subgroups , or combine conditions of O-pairs for maximal subgroups with seminormal, which prove the solvability or supersolvability of the group G. So our results generalize some given conclusions.