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Conjugacy class fusion from four maximal subgroups of the Monster

Anthony Pisani, Tomasz Popiel

TL;DR

This work computes the fusion of conjugacy classes from four specified maximal subgroups of the Monster group $\mathbf{M}$ into $\mathbf{M}$ itself, enabling their inclusion in the GAP Character Table Library. By constructing each subgroup inside the mmgroup framework and using a combination of centraliser-based character evaluation $\chi_{\mathbf{M}}$ and GAP’s PossibleClassFusions, the authors determine, for every subgroup class, the corresponding monster class. The results are presented as explicit fusion data in the respective theorems and are supported by reproducible computations documented in accompanying Python code. This provides concrete, machine-checkable fusion tables that enhance Monster character-theory computations and illustrate the viability of mmgroup for large sporadic-group problems.

Abstract

We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in question are $(\text{PSL}_2(11) {\times} \text{PSL}_2(11)){:}4$, $11^2{:}(5 {\times} 2\text{A}_5)$, $7^2{:}\text{SL}_2(7)$, and $\text{PSL}_2(19){:}2$. Our proofs are supported by reproducible calculations carried out using the Python package mmgroup, a computational construction of the Monster recently developed by Seysen.

Conjugacy class fusion from four maximal subgroups of the Monster

TL;DR

This work computes the fusion of conjugacy classes from four specified maximal subgroups of the Monster group into itself, enabling their inclusion in the GAP Character Table Library. By constructing each subgroup inside the mmgroup framework and using a combination of centraliser-based character evaluation and GAP’s PossibleClassFusions, the authors determine, for every subgroup class, the corresponding monster class. The results are presented as explicit fusion data in the respective theorems and are supported by reproducible computations documented in accompanying Python code. This provides concrete, machine-checkable fusion tables that enhance Monster character-theory computations and illustrate the viability of mmgroup for large sporadic-group problems.

Abstract

We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in question are , , , and . Our proofs are supported by reproducible calculations carried out using the Python package mmgroup, a computational construction of the Monster recently developed by Seysen.
Paper Structure (7 sections, 8 theorems, 1 equation, 2 tables)

This paper contains 7 sections, 8 theorems, 1 equation, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The maximal subgroup $({\rm PSL}_2(11) {\times} {\rm PSL}_2(11)){:}4$
  4. The maximal subgroup $11^2{:}(5 {\times} 2{\rm A}_5)$
  5. The maximal subgroup $7^2{:}\text{SL}_2(7)$
  6. The maximal subgroup ${\rm PSL}_2(19){:}2$
  7. Code listings

Key Result

Lemma 3.1

The elements $x_2,x_{11},x_4 \in \mathbf{M}$ in Listing fig:secL211 generate a maximal subgroup of $\mathbf{M}$ of shape $({\rm PSL}_2(11) {\times} {\rm PSL}_2(11)){:}4$.

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 5.1
  • proof
  • ...and 6 more