§3 — $|M_{23}|=10{,}200{,}960$

Proved

MathieuM23.m23_card

by Lucas · Oct 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

group-theorymathieu-group

The Mathieu group $M_{23}=\langle g_1,g_2\rangle\le S_{23}$ has order

|M_{23}|=10{,}200{,}960=2^7\cdot3^2\cdot5\cdot7\cdot11\cdot23.

This pins down that the explicit generators $g_1,g_2$ really generate the Mathieu group, and not a smaller or larger subgroup of $S_{23}$ .

Preamble

import Definitions.Def_MathieuM23_Group

Formal statement

namespace MathieuM23

theorem m23_card : Nat.card M23 = 10200960 := by sorry

end MathieuM23

Source

X. Huang, B. Jackson, K.-H. Lee, B. Poonen, R. Pries, S. Zhang, *The Mathieu group M23 is a Galois group over Q*, arXiv:2608.08538v1 (2026), https://arxiv.org/abs/2608.08538, p. 4, §3, first sentence

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Disclosure — NON-BLIND read-back. This read-back was written by the same agent that drafted the Lean statement (Aristotle, by Harmonic), with full knowledge of the source paper and of the intended meaning. It is not independent, blind testimony and must not be mistaken for an independent audit; a reviewer should compare it against the Lean code directly.

Statement. The subgroup $M_{23}$ of the permutation group of $\{0,\dots,22\}$ generated by the explicit permutations $g_1$ and $g_2$ has exactly $10200960$ elements. Cardinality is measured as a natural number; it would be $0$ for an infinite set, which cannot happen for a subgroup of a finite group. No hypotheses.

Human review

Endorsed by Shuze Chen · Oct 5, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 5, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

§3 — ∣M23∣=10,200,960|M_{23}|=10{,}200{,}960∣M23​∣=10,200,960

§3 — $|M_{23}|=10{,}200{,}960$