On SQ-Supplemented Subgroups of Finite Groups

Abstract

Let G be a finite group and H ≤ G. The subgroup H is called: S-permutable in G if HP = PH for all Sylow subgroups P of G; S-permutably embedded in G if each Sylow subgroup of H is also a Sylow subgroup of some S-permutable subgroup of G.

Let H be a subgroup of a group G. Then we say that H is SQ-supplemented in G if G has a subgroup T and an S-permutably embedded subgroup C ≤ H such that HT = G and T ∩ H ≤ C.

We study the structure of G under the assumption that some subgroups of G are SQ-supplemented in G. Some known results are generalized.

Key Words:

• Finite group
• Maximal subgroup
• p-Nilpotent group
• S-Permutable subgroup
• S-Permutably embedded subgroup
• Saturated formation
• Soluble group
• Supersoluble group
• SQ-Supplemented subgroup
• Weakly S-supplemented subgroup

2010 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20

ACKNOWLEDGMENT

The author is very grateful to the helpful suggestions of the referee.

Notes

Communicated by D. Macpherson.