Some design theoretic results on the conway group ·0

Abstract

Let Ω 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.