On an extension of Shlyk's theorem

TL;DR

This paper extends Shlyk's theorem by proving that in certain non-solvable groups, the intersection of specific maximal subgroups containing a Sylow normalizer is nilpotent, revealing new structural insights.

Contribution

It generalizes Shlyk's theorem to a broader class of non-solvable groups with specific subgroup intersection properties.

Findings

The intersection of all such maximal subgroups is nilpotent.

This intersection contains the normalizer of a Sylow subgroup.

The result applies to non-solvable groups with particular subgroup configurations.

Abstract

In this paper, we prove that the intersection of all non-nilpotent maximal subgroups of a non-solvable group containing the normalizer of some Sylow subgroup is nilpotent, which provides an extension of Shlyk's theorem.