arXiv subjects

Martin Seysen

Publications and source records attributed to Martin Seysen.

3 recordsLinked to original sources

arXiv2025

The Order of the Monster Finite Simple Group

We determine the order of the largest of the twenty-six sporadic simple groups known as the Monster, using a straightforward computational approach. The Monster is here defined as a subgroup of the symmetry group of the 196884-dimensional Griess algebra generated by a group of type $2^{1+24}_+.Co_1$ and an additional triality automorphism. Our approach is based on counting arguments for certain idempotents of the Griess algebra called axes. Our proof is self-contained, requiring only established properties of the Conway group as the automorphism group of the Leech lattice, and some of its subgroups. Although our approach is conceptually simple, it requires extensive calculation inside a 196884-dimensional matrix group that current computer algebra systems cannot easily handle directly. Instead, we use the software package mmgroup, developed by the second author, which supports fast calculations inside the Monster. To our knowledge, this paper contains the first self-contained computation of the order of the Monster. The Monster also acts on the Moonshine module V^#, which is a vertex operator algebra of central charge c=24. We provide a new proof that the Monster is the full automorphism group of the Griess algebra and of the Moonshine module using Borcherds' proof of the Monstrous Moonshine conjectures. In addition, we show that the Monster has exactly two conjugacy classes of involutions. The order of the Baby Monster, the second largest of the sporadic simple groups, is also determined.

math.GR↗

arXiv2022

A fast implementation of the Monster group

Let $\mathbb{M}$ be the Monster group, which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985 Conway has constructed a 196884-dimensional rational epresentation $\rho$ of $\mathbb{M}$ with matrix entries in $\mathbb{Z}[\frac{1}{2}]$. We describe a new and very fast algorithm for performing the group operation in $\mathbb{M}$. For an odd integer $p > 1$ let $\rho_p$ be the representation $\rho$ with matrix entries taken modulo $p$. We use a generating set $\Gamma$ of $\mathbb{M}$, such that the operation of a generator in $\Gamma$ on an element of $\rho_p$ can easily be computed. We construct a triple $(v_1, v^+, v^-)$ of elements of the module $\rho_{15}$, such that an unknown $g \in \mathbb{M}$ can be effectively computed as a word in $\Gamma$ from the images $(v_1 g, v^+ g, v^- g)$. Our new algorithm based on this idea multiplies two random elements of $\mathbb{M}$ in less than 30~milliseconds on a standard PC with an Intel i7-8750H CPU at 4 GHz. This is more than 100000 times faster than estimated by Wilson in 2013.

math.GR↗

arXiv2020

A computer-friendly construction of the monster

Let $\mathbb{M}$ be the monster group which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985, Conway has constructed a 196884-dimensional representation $\rho$ of $\mathbb{M}$ with matrix coefficients in $\mathbb{Z}[\frac{1}{2}]$. So these matrices may be reduced modulo any (not necessarily prime) odd number $p$, leading to representations of $\mathbb{M}$ in odd characteristic. The representation $\rho$ is based on representations of two maximal subgroups $G_{x0}$ and $N_0$ of $\mathbb{M}$. In ATLAS notation, $G_{x0}$ has structure $2_+^{1+24}.\mbox{Co}_1$ and $N_0$ has structure $2^{2+11+22}.( M_{24} \times S_3)$. Conway has constructed an explicit set of generators of $N_0$, but not of $G_{x0}$. This paper is essentially a rewrite of Conway's construction augmented by an explicit construction of an element of $G_{x0} \setminus N_0$. This gives us a complete set of generators of $\mathbb{M}$. It turns out that the matrices of all generators of $\mathbb{M}$ consist of monomial blocks, and of blocks which are essentially Hadamard matrices scaled by a negative power of two. Multiplication with such a generator can be programmed very efficiently if the modulus $p$ is of shape $2^k-1$. So this paper may be considered a as programmer's reference for Conway's construction of the monster group $\mathbb{M}$. We have implemented representations of $\mathbb{M}$ modulo 3, 7, 15, 31, 127, and 255.

math.GR↗