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Some necessary conditions for compatibility of groups

Zhaochen Ding, Gabriel Verret

TL;DR

This paper analyzes when two finite groups $L_1$ and $L_2$ can arise as quotients of a common group $G$ by isomorphic normal subgroups. It develops a framework based on characteristic functors, Melnikov formations, and Sims pairs to derive new necessary conditions for compatibility, including reductions via minimal witnesses and factor-structure arguments. A key contribution is proving that if $L_1$ and $L_2$ have no abelian composition factors, then they admit compatible normal series, with the proof built through Sims-pair reductions and radical considerations. These results yield practical criteria to prove non-compatibility and deepen the connection between Sims' Lemma and abstract (non-permutation) group compatibility, offering insight into the hierarchical structure of potential witnesses.

Abstract

Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. In this paper, we give new necessary conditions for two groups to be compatible.

Some necessary conditions for compatibility of groups

TL;DR

This paper analyzes when two finite groups and can arise as quotients of a common group by isomorphic normal subgroups. It develops a framework based on characteristic functors, Melnikov formations, and Sims pairs to derive new necessary conditions for compatibility, including reductions via minimal witnesses and factor-structure arguments. A key contribution is proving that if and have no abelian composition factors, then they admit compatible normal series, with the proof built through Sims-pair reductions and radical considerations. These results yield practical criteria to prove non-compatibility and deepen the connection between Sims' Lemma and abstract (non-permutation) group compatibility, offering insight into the hierarchical structure of potential witnesses.

Abstract

Two groups and are compatible if there exists a finite group with isomorphic normal subgroups and such that and . In this paper, we give new necessary conditions for two groups to be compatible.
Paper Structure (5 sections, 10 theorems, 21 equations)

This paper contains 5 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characteristic functors and minimal witness systems
  4. Sims' Lemma
  5. Main results

Key Result

Lemma 2.3

Let $\mathcal{K}$ be a Melnikov formation, let $G$ be a group, let $N,K\trianglelefteq G$ and let $f:G\rightarrow H$ be a homomorphism. The following statements hold.

Theorems & Definitions (27)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: Wielandt's Theorem wielandt1939Verallgemeinerung
  • Theorem 2.5: kurzweil2004theory
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 17 more