Thompson's sporadic group uniquely determined by the centralizer of a 2-central involution

In this article we give a self contained existence and uniqueness proof for that sporadic simple group which was discovered by J.G. Thompson [J.G. Thompson, A simple subgroup of E8 ( 3 ), in: N. Iwahori (Ed.), Finite Groups, Japan Soc. Promotion Science, Tokyo, 1976, pp. 113-116]. The centralizer H of a 2-central involution of that group has been described in terms of generators and relations by Havas, Soicher and Wilson in [G. Havas, L.H. Soicher, R.A. Wilson, A presentation for the Thompson sporadic simple group, in: W.M. Kantor, A. Seress (Eds.), Groups and Computation III, de Gruyter, Berlin, 2001, pp. 192-200]. Taking this presentation as the input of the second author's algorithm [G.O. Michler, On the construction of the finite simple groups with a given centralizer of a 2-central involution, J. Algebra 234 (2000) 668-693] we construct a simple subgroup G of GL248 ( 11 ) which has a 2-central involution z whose centralizer is isomorphic to H. In order to prove that the order of G is {Mathematical expression}, a faithful 143,127,000-dimensional permutation representation of this matrix group has been constructed on a supercomputer. In the second part of this article it is shown that any simple group G having a 2-central involution z with centralizer CG ( z ) ≅ H is isomorphic to G. We construct its concrete character table as well. © 2006 Elsevier Inc. All rights reserved.