ABSTRACT: Finite simple groups have even order. Therefore the centralizers H of involutions (elements of order 2) play an important role in their classification. By a well-known theorem of Brauer and Fowler, such a centralizer H can only be realized in a finite number of simple groups G . This is a remarkable result because there are several infinite series of simple groups which all have (up to isomorphism) the same Sylow 2-group. However, the Brauer-Fowler theorem is not constructive. It is therefore natural to ask whether one can calculate the orders of the target groups G from a given presentation of H in terms of generators and relations. In this lecture, a new group order formula will be presented which, together with a well-known group order formula of J. G. Thompson, provides a complete answer to this question.