Nearly m-embedded subgroups and solvability of finite groups

ABSTRACT

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G = HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.

KEYWORDS:

• Composition factor
• finite groups
• nearly m-embedded subgroup
• PSL(2,8)
• Sylow subgroups

AMS CLASSIFICATION:

• 20D10
• 20D20

Acknowledgments

The authors cordially thank the referee for detailed and valuable comments.