{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:30Z","timestamp":1753893810299,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/290","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:17:26Z","timestamp":1578716246000},"source":"Crossref","is-referenced-by-count":0,"title":["Some Design Theoretic Results on the Conway Group $\\cdot$0"],"prefix":"10.37236","volume":"17","author":[{"given":"Ben","family":"Fairbairn","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,1,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r18\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r18\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:01:52Z","timestamp":1579305712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/290","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,1,22]]},"article-number":"R18"}}