构造有限可解群的正规饱和群系
Constructing regular saturated formations of finite soluble groups
AI总结：
本文改进了Lucchini和Nemmi的结果，构造性地描述了有限可解群的饱和正规群系，证明其恰好为包含所有循环准素子群均为$K$-$\mathfrak{F}$-次正规子群的群的遗传群系，且该群系格与Steinitz格同构。
摘要：
对于有限群的群系 $\mathfrak{F}$，考虑一个图，其顶点为某个有限群的元素，且当且仅当两个顶点作为群的元素生成非 $\mathfrak{F}$-群时，它们由一条边连接。若在每一个群中，所述图的所有孤立顶点的集合与所有极大 $\mathfrak{F}$-子群的交集重合，则称遗传群系 $\mathfrak{F}$ 为正规的。本文获得了可解群的饱和正规群系的构造性描述，改进了 Lucchini 和 Nemmi 的结果。特别地，证明了可解群的饱和正规非空群系恰好是包含所有循环准素子群均为 $K$-$\mathfrak{F}$-次正规子群的群的可解群遗传群系 $\mathfrak{F}$。此外，还证明了可解群的饱和正规群系的格与 Steinitz 格是格同构的。
英文摘要：
For a formation $\mathfrak{F}$ of finite groups consider a graph whose vertices are elements of a finite group and two vertices are connected by an edge if and only if they generates non-$\mathfrak{F}$-group as elements of a group. A hereditary formation $\mathfrak{F}$ is called regular if the set of all isolated vertices of the described graph coincides with the intersection of all maximal $\mathfrak{F}$-subgroups in every group. The constructive description of saturated regular formations of soluble groups which improves the results of Lucchini and Nemmi is obtained in this paper. In particular, it is showed that saturated regular non-empty formations of soluble groups are just hereditary formations $\mathfrak{F}$ of soluble groups that contains every group all whose cyclic primary subgroups are $K$-$\mathfrak{F}$-subnormal. Also we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz's lattice.
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