On the construction of the finite simple groups with a given centralizer of a 2-central involution

Let H be a finite group having center Z(H) of even order. By the classical Brauer-Fowler theorem there can be only finitely many non-isomorphic simple groups G which contain a 2-central involution t for which C-G(t) congruent to H. In this article we give a deterministic algorithm constructing from the given group H all the finitely many simple groups G having an irreducib le p-modular representation M over some finite field F of odd characteristi c p > 0 with multiplicity-free semisimple restriction M-\H to H, if H satis fies certain natural conditions. As an application we obtain a uniform cons truction method for all the sporadic simple groups G not isomorphic to the smallest Mathieu group M-11. Furthermore, it provides a permutation represe ntation, and the character table of G. (C) 2000 Academic Press.