G-covering subgroup systems for the classes of finite soluble PST-groups

Abstract

A finite group G is called a PST-group (respectively, PT-group, T-group) if every subnormal subgroup of G is Sylow permutable (respectively, permutable, normal) in G. Let $\mathfrak{F}$ be a class of group and G a finite group. Then, a set Σ of subgroups of G is called a G-covering subgroup system for the class $\mathfrak{F}$ if $G\in \mathfrak{F}$ whenever $\Sigma \subseteq \mathfrak{F}.$ We prove that: (i) If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then, $\Sigma$ is a G-covering subgroup system for the class of all soluble PST-groups; (ii) if Σ is the set of all two-generated subgroups of G, then, $\Sigma$ is a G-covering subgroup system for the classes of all soluble PST-groups, all soluble PT-groups, and all soluble T-groups. We use these results to prove the following characterizations of soluble PT-groups and T-groups: Suppose that a set of subgroups Σ contains at least one supplement to each maximal subgroup of every Sylow subgroup of G. Then, G is a soluble PT-group (respectively, a soluble T-group) if and only if every subgroup in Σ is a soluble PT-group (respectively, a soluble T-group) and at least one of the nonidentity Sylow subgroups of G is an Iwasawa (respectively, a Dedekind) group.

Keywords:

• Finite group
• G-covering subgroup system
• PST-group
• PT-group
• Sylow permutable subgroup

2020 Mathematics Subject Classification:

• 20D10

Additional information

Funding

Research was supported by the Hainan Provincial Natural Science Foundation of China (Grant No. 119MS002) and the NNSF of China (No. 11771409).