A New Approach from Lattice of Subgroup Sets to Generalized Solvable Extension Formations
Ran Li, Long Miao, Wenxia Zhou, Yinan Chen
TL;DR
The paper develops a framework to decompose morphisms from the lattice of maximal subgroup sets to generalized solvable extension formations by leveraging maximal-subgroup functors, local and generating morphisms, and contraction/extension functors. It constructs generalized solvable extension formations via formation functions on subsets of finite simple groups, introducing classes such as $\mathfrak{J}_{pr}$, $\mathfrak{J}$, $\mathfrak{F}'$, and $\mathfrak{F}''$, together with their extensions and localizations. Localization via $\mathcal L_1$, $\mathcal L_2$, and related functors is developed to constrain groups into target formations, with inductive proofs showing how emptiness conditions on localized sets imply solvable or near-solvable structure. Additionally, the work analyzes the order-reversing morphism $g$ linking $MAX_2(G)$ to formations, elucidating how lattice operations correspond to formation operations and enriching the understanding of local-global behavior in finite groups.
Abstract
In this paper, we establish the decomposition of morphisms from lattice of subgroup sets to generalized solvable extension formations. To achieve this, we develop a unified framework involving maximal subgroup functors, generating formation morphism and contraction-extension functors. In particular, solvability-induced sets of maximal subgroups are determined and generating formation morphism gives rise to generalized solvable extension formations.