§3 — $M_{23}$ is 4-transitive on 23 points

Proved

MathieuM23.m23_four_transitive

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

group-theorymathieu-grouppermutation-groups

The natural action of $M_{23}$ on the $23$ points is $4$ -transitive: for any two ordered $4$ -tuples $(x_1,\dots,x_4)$ and $(y_1,\dots,y_4)$ of pairwise distinct points there is $\sigma\in M_{23}$ with $\sigma(x_i)=y_i$ for $i=1,\dots,4$ .

Preamble

import Definitions.Def_MathieuM23_Group

Formal statement

namespace MathieuM23

theorem m23_four_transitive : MulAction.IsMultiplyPretransitive M23 (Fin 23) 4 := 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, second 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. Consider the action of $M_{23}=\langle g_1,g_2\rangle$ on $\{0,\dots,22\}$ by evaluation, $\sigma\cdot x=\sigma(x)$ . This action is $4$ -pretransitive. For every two injective maps $x,y:\{1,2,3,4\}\to\{0,\dots,22\}$ (ordered quadruples of pairwise distinct points) there is $\sigma\in M_{23}$ with $\sigma(x_i)=y_i$ for all $i$ . No hypotheses. ( $4$ -pretransitivity is non-vacuous here since $23\ge4$ .)

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 — M23M_{23}M23​ is 4-transitive on 23 points

§3 — $M_{23}$ is 4-transitive on 23 points