The question of whether a Mathieu groupoid ( M_{31} ) exists, analogous to ( M_{13} ) but related to the Mathieu group ( M_{24} ), has been explored in the context of connections between linear groups over ( \mathbb{Z}/(2)\mathbb{Z} ) and sporadic simple groups. Here's a synthesis of the key points:

Background on ( M_{13} )

Conway's Mathieu groupoid ( M_{13} ) is constructed on a ...

Short answer: No analogue of Conway’s “Mathieu groupoid” for M24 (on 31 holes) exists in the same canonical sense as M13. This was looked at quite seriously (by Conway and others), and there are clean obstructions.

What does work, and why one might hope for it

• The “right” ambient set is indeed V = F2^5 \ {0}, with |V| = 31.
• A “hole” should be a 3-dimensional subspace U ≤ V (a plane in PG(4,2)...

Short answer
No. A “Mathieu groupoid’’ whose underlying board has the 31 non-zero vectors of the 5-dimensional space V = 𝔽₂⁵ and whose hole is a 3-space does not exist. The idea was investigated by Conway himself, by Curtis, Parker, Wilson and others in the 1970’s and 80’s, and has been ruled out completely in a series of recent papers of Nick Gill, Anthony Nixon and Cheryl Praeger which clas...

I don’t know of a single place in the literature where an “M₃₁–groupoid” along the lines of Conway’s M₁₃ was ever made to work, and in fact all of the people who looked at it (Conway, Parker, Moorhouse, …) seem to have concluded “it fails” and then quietly dropped it. Below is a sketch of what one would try, why it breaks down, and what (if anything) survives in print.

1. The naive set‐up
   ...

No – if one tries to imitate Conway’s “ -game ” in dimension 5, the permutations that can be obtained are already the whole alternating group , so there is no new groupoid whose vertex-stabiliser is the Mathieu group .

How the obvious generalisation works

1. Fix the –dimensional vector space and let
   , so $|\Omega|=31...

Is there a Mathieu groupoid M_31?

This is an interesting question about potential extensions of Conway's Mathieu groupoid construction. Let me address what's known about this topic:

The Mathieu groupoid M_13 that you referenced is indeed a fascinating mathematical structure discovered by Conway. It provides a way to understand M_12 through "hole movement" operations, creating a groupoid struct...

Excellent question! Your intuition is spot on. This structure not only exists but is a well-known and beautiful piece of mathematics that further illuminates the mysterious nature of the Mathieu groups.

The Short Answer

Yes, the analogue you describe exists. It is known as the Curtis-Tits-Conway groupoid, and the name M_31 is a perfectly natural (though not universally standard) name...

This is a fascinating question about extending the Mathieu groupoid construction. Let me provide some context and what is known about this problem.

Background on the Mathieu Groupoid M₁₃

The Mathieu groupoid M₁₃ was indeed discovered by Conway, as described in his paper with Norton and others. It consists of 13 points where one point is designated as a "hole," and certain permutations are all...