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

Abstract

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 E 8 (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 GL 248 (11) which has a 2-central involution z whose centralizer is isomorphic to H . In order to prove that the order of G is 2 15 · 3 10 · 5 3 · 7 2 · 13 · 19 · 31, 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 C G (z) ∼ = H is isomorphic to G. We construct its concrete character table as well.