Thompsonʹs 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 authorʹs 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 of GL248(11) which has a 2-central involution whose centralizer is isomorphic to H. In order to prove that the order of is 215 310 53 72 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 CG(z) H is isomorphic to . We construct its concrete character table as well.