Simultaneous Constructions of the Sporadic Groups Co_2 and Fi_{22}

TL;DR

This paper provides self-contained proofs for the existence of the sporadic groups Co_2 and Fi_{22} using an algorithm that constructs these groups from specific irreducible subgroups over GF(2), including character tables and presentations.

Contribution

It introduces a new constructive method to realize Co_2 and Fi_{22} from irreducible subgroups, offering an alternative to their original discovery methods.

Findings

Constructed Co_2 inside GL_{23}(13)

Constructed Fi_{22} inside GL_{78}(13)

Calculated character tables and presentations for both groups

Abstract

In this article we give self-contained existence proofs for the sporadic simple groups Co_2 and Fi_{22} using the second author's algorithm [10] constructing finite simple groups from irreducible subgroups of GL_n(2). These two sporadic groups were originally discovered by J. Conway [4] and B. Fischer [7], respectively, by means of completely different and unrelated methods. In this article n=10 and the irreducible subgroups are the Mathieu group M_{22} and its automorphism group Aut(M_{22}). We construct their five non-isomorphic extensions E_i by the two 10-dimensional non-isomorphic simple modules of M_{22} and by the two 10-dimensional simple modules of A_{22} = Aut(M_{22}) over F=GF(2). In two cases we construct the centralizer H_i = C_{G_i}(z_i) of a 2-central involution z_i of E_i in any target simple group G_i. Then we prove that all the conditions of Algorithm 7.4.8 of [11] are satisfied. This allows us to construct G_3 $\cong$ Co_2 inside GL_{23}(13) and G_2 $\cong$ Fi_{22} inside GL_{78}(13). We also calculate their character tables and presentations.