Year: 2026
Type: preprint
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. This deposit contains the unchanged 55-page PDF submitted to arXiv on September 27, 2026, as submission 8137031 (math.GR, math.CO, math.RA). The PDF bears the arXiv submission stamp. The original arXiv submission uses the arXiv.org perpetual, non-exclusive distribution license.
Source: arXiv (Cornell University)
Author: Höhn, Gerald
Institution: Kansas State University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214555775