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 CG(t)≅H. In this article we give a deterministic algorithm constructing from the given group H all the finitely many simple groups G having an irreducible p-modular representation M over some finite field F of odd characteristic p>0 with multiplicity-free semisimple restriction M|H to H, if H satisfies certain natural conditions. As an application we obtain a uniform construction method for all the sporadic simple groups G not isomorphic to the smallest Mathieu group M11. Furthermore, it provides a permutation representation, and the character table of G.