Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Some design theoretic results on the Conway group ·0. | Some design theoretic results on the Conway group -0 Ben Fairbairn School of Mathematics University of Birmingham Birmingham B15 2TT United Kingdom f airbaib@ Submitted Oct 3 2008 Accepted Jan 14 2010 Published Jan 22 2010 Mathematics Subject Classification 05B99 05E10 05E15 05E18 Abstract Let Q 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 c Q be subsets of size four tetrads . The structure of S forces each tetrad to define a certain partition of Q 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 AuB 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. 1 Introduction The Leech lattice A was discovered by Leech in 1965 in connection with the packing of spheres into 24-dimensional space R24 so that their centres form a lattice. Its construction relies heavily on the rich combinatorial structure of the Mathieu group M24. Leech himself considered the group of symmetries fixing the origin 0 he had enough geometric evidence to predict the order of this group to within a factor of two but could not prove the existence of all the symmetries he anticipated. John Conway subsequently produced a beautifully simple additional symmetry of A and in doing so determined the order of the group it generated together with the monomial subgroup used in the .