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{\bf Ben Fairbairn}
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{\bf Some Design Theoretic Results on the Conway Group $\cdot$0}
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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.
\bye