Digital Finite Group Theory

Abstract

This book develops a digital formulation of finite group theory, in which every finite group of order $N$ is represented as the alphabet $\{0,1,\dots,N-1\}$ endowed with the transported group product. Multiplication, inversion and related structure are written explicitly as integer operations on labels. The foundations treat digit functions, codes of permutations and $r$-ary operators, finite magmas, and constructions of laws from diagrams. Classical parts of finite group theory, including conjugacy classes, stabiliser chains and characters, are recast as identities between digits of codes. The book gives explicit digital laws for all groups of order at most 32, the finite quotients of the 17 plane and 230 space crystallographic groups, the symmetric and alternating groups, the finite simple groups of Lie type, and all 26 sporadic groups. For most sporadic groups the laws are obtained by chain constructions; for the Baby Monster and the Monster they are computed structurally from the 196884-dimensional representation and evaluated with the mmgroup package. The accompanying software archive reproduces the laws, tables and worked examples as verifiable integer data.