arXiv subjects

Anthony Pisani

Publications and source records attributed to Anthony Pisani.

5 recordsLinked to original sources

arXiv2026

Finding 59:29 in the Monster

The classification of the maximal subgroups of the Monster group has been completed recently, and explicit generators for each such subgroup (up to conjugacy) have been made available for the software mmgroup, with the exception of the maximal subgroup 59:29, see Dietrich et al. (Adv. Math., 2025; J Algebra, 2026). We provide explicit generators for this last maximal subgroup and comment on the extensive search that led to finding them. Our method is similar to, but significantly more involved than Bray et al.'s (London Math. Soc. J. Comput. Math., 2016) approach for constructing 47:23 in the Baby Monster. Our result allows us to provide a new short proof that the Monster does not have a subgroup PSL2(59), correcting a result of Holmes and Wilson (J. London Math. Soc., 2004).

math.GR↗

arXiv2025

The Saxl hypergraph of a permutation group

Given a permutation group $G \le \mathrm{Sym}(\Omega)$, a subset $B$ of $\Omega$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$, Burness and Giudici define the Saxl graph of $G$ to be the graph on $\Omega$ with bases of size 2 as edges. Later work of Freedman et al. extends this notion to any group for which $b(G) \ge 2$, taking the pairs of points contained in bases of size $b(G)$ for edges. We study an alternative generalisation, the Saxl hypergraph, where bases of size $b(G)$ are themselves the edges. In particular, we consider groups with complete Saxl hypergraphs, primitive groups whose Saxl hypergraphs have flag-spanning tours, and appropriate generalisations of Burness and Giudici's Common Neighbour Conjecture.

math.GR↗

arXiv2025

Computing the Character Table of a 2-local Maximal Subgroup of the Monster

We employ the recently developed hybrid and mmgroup computational models for groups to calculate the character table of $N(\rm{2B}^5) \cong 2^{5+10+20}.( \rm{S}_3 \times \rm{L}_5 {2} )$, a maximal subgroup of the Monster sporadic simple group. This completes the list of the character tables of maximal subgroups of the Monster. Our approach illustrates how the aforementioned computational models can be used to calculate relatively straightforwardly in the Monster.

math.GR↗

arXiv2024

Explicit construction of the maximal subgroups of the Monster

Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$.

math.GR↗

arXiv2024

Conjugacy class fusion from four maximal subgroups of the Monster

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.

math.GR↗