Generalized Frattini subgroup and supplements of subgroups of groups

Abstract

Let ${\Phi }_a(G)$ and ${\Phi }_n(G)$ denote the intersection of all abnormal and normal maximal subgroups of G, respectively. S is called ${\Phi }_a$(or ${\Phi }_n$)-supplemented in G if there exists a proper normal subgroup N of G such that G = SN and $S\cap N\le {\Phi }_a(S)$(or ${\Phi }_n(S)$). In this paper we characterize many properties of groups by using ${\Phi }_a$ and ${\Phi }_n$-supplementation.

Communicated by Sudarshan Kumar Sehgal

Keywords:

• Generalized Frattini subgroup
• solvable group
• supplement of subgroup

AMS Subject Classifications:

• 20D15
• 20E34
• 20F18

Acknowledgments

The authors are grateful to the referee. Many corrections were given by the referee.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11871360).