Subrack lattice determines the derived length

TL;DR

The paper investigates whether the derived length of a finite solvable group is determined by its subrack lattice. It develops a lattice-based framework using $\mathsf{A}(G)$ to locate maximal normal abelian subgroups and transfers coset structure through a lattice isomorphism to mirror a chain of abelian quotients in any isomorphic partner group. It proves that if $G$ is solvable with derived length $\ell$ and $\mathcal{R}(G)\cong \mathcal{R}(H)$, then $H$ is solvable with the same derived length $\ell$, and the approach extends to determining the nilpotence class via centers. This establishes a strong combinatorial invariant: the derived length (and related nilpotence data) are encoded in the subrack lattice, linking rack theory to classical group-theoretic structure and enabling lattice-based classification.

Abstract

We prove the following: If $G$ is a finite solvable group and $H$ is another group whose subrack lattice is isomorphic to the subrack lattice of $G$, then $H$ is a solvable group and the derived length of $H$ coincides with the derived length of $G$.

Theorems & Definitions (8)

• Theorem 1: see HSW19
• Theorem 2
• Lemma 3: see Kay21
• Lemma 4: see Kay21
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Remark