Structure of abstract finite simple groups

Gerhard Michler, University of Essen

By the celebrated Brauer-Fowler Theorem there are only finite simple groups G having an involution z such that its centralizer C_G(z) is isomorphic to a given group H. This is an important observation because there are several series of infinitely many simple groups with isomorphic Sylow 2-subgroups of constant order. In this lecture I will present a structure theorem for an abstract finite simple group G. It is the theoretical basis for an algorithm constructing the simple target groups G from a presentation of the given group H. Furthermore, I will present a new group order formula for simple groups with a unique conjugacy class of involutions. The structure theorem, Thompson's group order formula and my group order formula allow the calculation of the orders of the target groups G by using only data which can be derived from the presentation of H. Successful applications of these results like the uniqueness proof by Weller, Previtali and myself of the long standing problem of the uniqueness of the sporadic Thompson group will be mentioned. Furthermore, some open problems in the theory of finite simple groups will be stated.