Some Design Theoretic Results on the Conway Group $\cdot$0

Let $\Omega$ be a set of 24 points with the structure of the (5,8,24) Steiner system, $\cal{S}$, defined on it. The automorphism group of $\cal{S}$ acts on the famous Leech lattice, as does the binary Golay code defined by $\cal{S}$. Let $A,B\subset\Omega$ be subsets of size four ("tetrads"). The structure of $\cal{S}$ forces each tetrad to define a certain partition of $\Omega$ 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 $\zeta_A$. It is well known that for $\zeta_A$ and $\zeta_B$ 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 $\zeta_A$ and $\zeta_B$ will commute if and only if $A\cup B$ is contained in a block of $\cal{S}$. 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.