2020/02/22 by Martin Seysen, Seysen, Martin · 2 citations
Computer Science · Engineering · Mathematics · #20C11 (Secondary) #20C34 (Primary) 20D08 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2002.10921
openalex publication_date 2020/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbM 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 ρ of \mathbbM with matrix coefficients in ℤ[(1)/(2)]. So these matrices may be reduced modulo any (not necessarily prime) odd number p, leading to representations of \mathbbM in odd characteristic. The representation ρ is based on representations of two maximal subgroups Gx0 and N0 of \mathbbM. In ATLAS notation, Gx0 has structure 2+1+24.Co1 and N0 has structure 22+11+22.( M24 × S3). Conway has constructed an explicit set of generators of N0, but not of Gx0. This paper is essentially a rewrite of Conway's construction augmented by an explicit construction of an element of Gx0 ∖ N0. This gives us a complete set of generators of \mathbbM. It turns out that the matrices of all generators of \mathbbM 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 2k-1. So this paper may be considered a as programmer's reference for Conway's construction of the monster group \mathbbM. We have implemented representations of \mathbbM modulo 3, 7, 15, 31, 127, and 255.