New criteria for the solvability of finite groups

On solvability of finite groups with some ss-supplemented subgroups

“…More precisely, we improve and generalize the results of Hall [9], Arad and Ward [1], BallesterBolinches et al [3], Asaad and Ramadan [2], Guo et al [7], [16], and Heliel [11] …”

“…Conversely, assume that the Sylow 2-subgroups and Sylow 3-subgroups of G are ss-supplemented in G. With the same argument as in the proof of Theorem 1.5, we know that the Sylow 2-subgroups and Sylow 3-subgroups of G are complemented in G. By Arad and Ward in [1], G is solvable as desired.…”

“…In the literature, many authors have investigated the structure of the group G under the assumption that some subgroups of G are well-situated in G. For example, Hall in [9] proved that a group G is solvable if and only if each Sylow subgroup of G is complemented in G. Arad and Ward in [1] obtained a nice generalization of Hall's theorem. In fact, they proved that a group G is solvable if the Sylow 2-subgroups and Sylow 3-subgroups of G are complemented in G. Moreover, Hall in [10] proved that a group G is supersolvable with elementary abelian Sylow subgroups if and only if every subgroup of G is complemented in G. Ballester-Bolinches and Guo in [4] analysed the class of groups for which every subgroup of prime order is complemented.…”

On inequalities of subgroups and the structure of finite groups

“…All authors read and approved the final manuscript. 1 Department of Mathematics, Chongqing University of Arts and Science, Chongqing, 402160, P.R. China.…”

On complemented subgroups of finite groups

“…It is obvious that the existence of supplements for some families of subgroups of a group gives a lot of information about its structure. For instance, Kegel [8], [9] showed that a group G is soluble if every maximal subgroup of G either has a cyclic supplement in G or if some nilpotent subgroup of G has a nilpotent supplement in G. Hall [6] proved that a group G is soluble if and only if every Sylow subgroup of G is complemented in G. Arad and Ward [1] proved that a group G is soluble if and only if every Sylow 2-subgroup and every Sylow 3-subgroup of G are complemented in G. More recently, A. Ballester-Bolinches and Guo Xiuyun [2] proved that the class of all finite supersoluble groups with elementary abelian Sylow subgroups is just the class of all finite groups for which every minimal subgroup is complemented.…”