The Leech lattice Λ and the Conway group ⋅𝑂 revisited

Abstract

We give a new, concise definition of the Conway group ⋅ O \cdot \mathrm {O} as follows. The Mathieu group M 24 \mathrm {M}_{24} acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of ( 24 4 ) \left ({24}\atop {4} \right ) tetrads. We use this action to define a progenitor P P of shape 2 ⋆ ( 24 4 ) : M 24 2^{\star \left ( 24 \atop 4 \right )}:\mathrm {M}_{24} ; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P P by this relator results in ⋅ O \cdot \mathrm {O} . Consideration of the lowest dimension in which ⋅ O \cdot \mathrm {O} can act faithfully produces Conway’s elements ξ T \xi _T and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ⋅ O \cdot \mathrm {O} of the integral vectors in R 24 {\mathbb R}_{24} .