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Some design theoretic results on the Conway group ·0

By Ben Fairbairn


Let Ω be a set of 24 points with the structure of the (5,8,24) Steiner system, S, defined on it. The automorphism group of S acts on the famous Leech lattice, as does the binary Golay code defined by S. Let A,B ⊂ Ω be subsets of size four (“tetrads”). The structure of S forces each tetrad to define a certain partition of Ω into six tetrads called a sextet. For each tetrad Conway defined a certain automorphism of the Leech lattice that extends the group generated by the above to the full automorphism group of the lattice. For the tetrad A he denoted this automorphism ζA. It is well known that for ζA and ζB to commute it is sufficient to have A and B belong to the same sextet. We extend this to a much less obvious necessary and sufficient condition, namely ζA and ζB will commute if and only if A∪B is contained in a block of S. We go on to extend this result to similar conditions for other elements of the group and show how neatly these results restrict to certain important subgroups.

Year: 2010
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