The Conway-Parker algebra and the largest Fischer group
Abstract
We give a direct, self-contained construction of the three sporadic Fischer groups Fi24', Fi23, and Fi22 from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group Fi24. Its commutator subgroup gives Fi24', while Fi23 and Fi22 arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and M24. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.
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BibTeX
@misc{hohn2026conwayparker,
title = {{The Conway-Parker algebra and the largest Fischer group}},
author = {Gerald Höhn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33331},
eprint = {2609.33331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33331},
}
RIS
TY - UNPB
TI - The Conway-Parker algebra and the largest Fischer group
AU - Gerald Höhn
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33331
UR - https://arxiv.org/abs/2609.33331
AB - We give a direct, self-contained construction of the three sporadic Fischer groups Fi_24', Fi_23, and Fi_22 from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group Fi_24. Its commutator subgroup gives Fi_24', while Fi_23 and Fi_22 arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and M_24. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.
ER -