# Some design theoretic results on the Conway group ·0

Fairbairn, Ben
(2010)
Some design theoretic results on the Conway group ·0.
*Electronic Journal of Combinatorics* 17
(R18),
pp. 1-11.
ISSN 1077-8926.

Text (Refereed)
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## Abstract

Let Om 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 ⊂ L be subsets of size four (“tetrads”). The structure of S forces each tetrad to define a certain partition of 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 A∪B 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.

## Metadata

Item Type: | Article |
---|---|

School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |

Depositing User: | Ben Fairbairn |

Date Deposited: | 20 Dec 2012 13:33 |

Last Modified: | 09 Aug 2023 12:32 |

URI: | https://eprints.bbk.ac.uk/id/eprint/5434 |

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