Construction of Co_1 from an irreducible subgroup M_{24} of GL_{11}(2)

TL;DR

This paper provides a self-contained proof of the existence of the sporadic simple group Co_1 by constructing it from an irreducible subgroup of GL_{11}(2) and verifying conditions of a specific algorithm.

Contribution

It introduces a novel construction of Co_1 using an irreducible subgroup of GL_{11}(2) and an algorithmic approach to prove its properties.

Findings

Successfully constructed Co_1 from M_{24} and an 11-dimensional GF(2) module.

Verified all algorithmic conditions to confirm the group's isomorphism with Co_1.

Provided a faithful permutation representation and used known presentations for validation.

Abstract

In this article we give an self contained existence proof for J. Conway's sporadic simple group Co_1 [4] using the second author's algorithm [14] constructing finite simple groups from irreducible subgroups of GL_n(2). Here n = 11 and the irreducible subgroup is the Mathieu group M_{24}. From the split extension E of M_{24} by a uniquely determined 11-dimensional GF(2)M_{24}-module V we construct the centralizer H = C_G(z) of a 2-central involution z of E in an unknown target group G. Then we prove that all the conditions of Algorithm 2.5 of [14] are satisfied. This allows us to construct a simple subgroup G of GL_{276}(23) which we prove to be isomorphic with Conway's original sporadic simple group Co_1 by means of a constructed faithful permutation representation of G and Soicher's presentation [16] of the original Conway group Co_1.