The Conway-Wales lattice and the Rudvalis and Tits groups
Abstract
We construct the Conway-Wales lattice from a self-dual even code C over ℤ4 whose binary layers interweave the Hamming codes ℋ7 and ℋ8. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order 583 704 576 000. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order 17 971 200. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.
Related
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BibTeX
@misc{hohn2026conwaywales,
title = {{The Conway-Wales lattice and the Rudvalis and Tits groups}},
author = {Gerald Höhn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10553},
eprint = {2610.10553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10553},
}
RIS
TY - UNPB
TI - The Conway-Wales lattice and the Rudvalis and Tits groups
AU - Gerald Höhn
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.10553
UR - https://arxiv.org/abs/2610.10553
AB - We construct the Conway-Wales lattice from a self-dual even code C over ℤ_4 whose binary layers interweave the Hamming codes ℋ_7 and ℋ_8. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order 583 704 576 000. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order 17 971 200. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.
ER -