Sách Some design theoretic results on the Conway group

Abstract
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
our (“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
o 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
o 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
or other elements of the group and show how neatly these results restrict to certain
mportant subgroups.