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Báo cáo toán học: "Some design theoretic results on the Conway group ·0"

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@maths.bham.ac.uk 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 .