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.

摘要

有限群G ^称为PST -基（分别为PT -基，Ť -基）如果每次正规子群G ^是西洛置换在（分别为，置换，正常）ģ。让$\mathfrak{F}$是一个群，G是一个有限群。然后，G的子群的集合 Σ称为类的G覆盖子群系统$\mathfrak{F}$ 如果 $G\in \mathfrak{F}$ 每当 $\Sigma \subseteq \mathfrak{F}.$我们证明了：（i）。如果组亚组Σ的摹包含至少一个补充的每一个西洛子群的每个极大子群摹，然后，$\Sigma$是所有可溶PST群类的G覆盖子群系统；(ii) 如果Σ是G的所有两个生成子群的集合，则，$\Sigma$是所有可溶性 PST 基团、所有可溶性PT基团和所有可溶性T基团的G覆盖子群系统。我们使用这些结果来证明可溶PT 群和T群的以下特征： 假设一组子群 Σ 包含对G的每个 Sylow 子群的每个最大子群的至少一个补充。那么，G是一个可溶的 PT-群（分别是一个可溶的T -群）当且仅当 Σ 中的每个子群都是一个可溶的PT -群（分别是一个可溶的T -群）和至少一个非同一性 Sylow 子群的G 是 Iwasawa（分别是 Dedekind）组。