In this series of lectures I will give a survey on the classification problem of finite simple groups. In particular, I will describe R. Brauer’s requirements for a classification scheme. Furthermore, I will address his statement that it may be impossible to classify the isomorphism classes of finite simple groups.

First I will describe the Brauer-Fowler theorem which states that there are only finitely many simple groups G which have an involution z in the center of a Sylow 2-subgroup such that C_G(z) is isomorphic to a given group H of even order. Two non isomorphic simple groups G_1 and G_2 with a given centralizer H are called satellites of each other.

I will give a survey about the famous classification theorems of the simple groups G having a dihedral, semi-dihedral or a strongly embedded subgroup.

My structure theorem says that a Sylow 2-subgroup S of any other simple group G contains a non cyclic elementary abelian characteristic subgroup A. Throughout the remainder of this abstract I only discuss this open case of the classification under the additional hypothesis that there is such a subgroup A such that C_G(A) is a 2-group. This hypothesis is satisfied in all known sporadic simple groups different from M_11 which has a semi-dihedral Sylow 2-subgroup.

I will describe and demonstrate explicit applications of my algorithm constructing all the satellites from a given centralizer H.

The main problem of the classification is the question: How does one find a centralizer H of a simple group? For that problem I found another algorithm constructing such groups H from indecomposable subgroups T of GL_n(2). This algorithm will be explained. Several explicit applications will be given. In particular, all Conway, Fischer and Janko simple groups have been constructed this way by my former students Weller, Kim and myself. But also some classical and alternating groups have been constructed so.

I will also address the question: Is there a 27th sporadic simple group? I refer to a suggestion of an article of J. G. Thompson. It yields a technically demanding problem for my algorithm.

Furthermore, I will present D. G. Higman’s theorem on the infinite representation type of the simple groups. L. Wang’s and my study of the existence and uniqueness of the Tits group shows that one has to study indecomposable subgroups and not only irreducible subgroups of GL_n(2) in an systematic search for simple groups.