On c-Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ℱ.

Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ℱ.

Key Words:

• c-Normal subgroup
• Supersolvable group
• The generalized Fitting subgroup
• Formation

2000 Mathematics Subject Classification:

• 20D10
• 20D20

Acknowledgments

The authors would like to thank T. R. Berger for his helpful discussion during his sabbatical visiting. The authors would also like to thank Dejian Lai for his help in correcting grammatical inconsistencies.

The authors are supported in part by NSFG, Fund from Education Ministry and AC-ZSU.