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The Leech lattice $ \Lambda$ and the Conway group $ \cdot {O}$ revisited

Authors: John N. Bray and Robert T. Curtis
Journal: Trans. Amer. Math. Soc. 362 (2010), 1351-1369
MSC (2000): Primary 20D08; Secondary 20F05
Published electronically: October 20, 2009
MathSciNet review: 2563732
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Abstract: We give a new, concise definition of the Conway group $ \cdot {O}$ as follows. The Mathieu group $ \mathrm{M}_{24}$ acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $ \left({24}\atop{4} \right)$ tetrads. We use this action to define a progenitor $ P$ of shape $ 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$ by this relator results in $ \cdot {O}$. Consideration of the lowest dimension in which $ \cdot {O}$ can act faithfully produces Conway's elements $ \xi_T$ and the 24-dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under $ \cdot {O}$ of the integral vectors in $ {\mathbb{R}}_{24}$.

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Additional Information

John N. Bray
Affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, United Kingdom

Robert T. Curtis
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

Keywords: Conway group, symmetric generation
Received by editor(s): November 23, 2007
Received by editor(s) in revised form: January 7, 2008
Published electronically: October 20, 2009
Dedicated: Dedicated to John Horton Conway as he approaches his seventieth birthday.
Article copyright: © Copyright 2009 by the authors

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