Some design theoretic results on the Conway group ·0

Fairbairn, Ben (2010) Some design theoretic results on the Conway group ·0. Electronic Journal of Combinatorics 17 (R18), pp. 1-11. ISSN 1077-8926.

Abstract

Let Om 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 ⊂ L be subsets of size four (“tetrads”). The structure of S forces each tetrad to define a certain partition of 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 ZA. It is well known that for ZA and ZB 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 ZA and ZB 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.

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