English translation of [72]
by Robert E. Molzon and Peter M. Neumann:
Helmut Wielandt's Acceptance of Membership of the Heidelberger Akademie der Wissenschaften
For a personal reason I feel an obligation to offer especial thanks to the Academy for the honour of election to full membership; namely, it gives me a direct connection with a city which several of my forefathers served. One of them was District Surveyor a hundred years ago. Perhaps it was from him that my mother had her bent for mathematics, which she then passed on to me.
Born the son of the pastor of a village of the Markgräfler Land, I grew up in Berlin.
The steps of my scientific career are quickly recounted: studies in mathematics and physics at Berlin University, the doctorate in 1935, membership of the editorial staff of Jahrbuch über die Fortschritte der Mathematik, an assistant's position and Habilitation at Tübingen; work during the second half of the war in the Aero- dynamics Research Institute at Göttingen, an associate professorship in mathematics in Mainz 1946, a professorship in Tübingen in 1951.
The most enduring impressions of my studies I owe to the lectures of Erhard Schmidt and Issai Schur. It is my perhaps unattainable ambition to combine their lecturing styles. While Schmidt gave his listeners the feeling that they were directly involved, experiencing with him the tentative development of a direction of mathe- matical thinking arising from a problem, Schur presented crystal clear finished theories polished to the last detail.
It is to one of Schur's seminars that I owe the stimulus to work with permutation groups, my first research area. At that time the theory had nearly died out. It had developed last century, but at about the turn of the century had been so completely superseded by the more generally applicable theory of abstract groups that by 1930 even important results were practically forgotten—to my mind unjustly. Even today permutation groups are still a powerful tool for describing and studying the sym- metries of finite systems, and they also fulfill a well-defined role within the scope of the abstract theory of finite groups. Since my dissertation I have returned time and again to this fascinating area of inquiry, where the questions are closely interwoven with number theory, combinatorics and finite geometries; it is now reviving also in other countries.
The work on permutation groups led me inevitably to involvement with the struc- ture theory of finite groups. In the twenties this theory had also fallen into neglect for reasons which I shall go into later. But Philip Hall's fundamental papers had already
XIV Helmut Wielandt's Acceptance of Membership
revitalised it. Where Hall had started from arithmetical questions and product decompositions, my own work was triggered by a question of Robert Remak of a quite different type: is the group generated by two subgroups that occur in com- position series always of the same kind? In my Habilitationsschrift I expanded the discovery that this question can be answered affirmatively to a detailed study of the normal structure of finite groups. It was much later that surprisingly close relation- ships with arithmetic structure and product decompositions emerged. The resulting connection with Hall's theory opens up promising problem areas.
The group-theoretic work was interrupted for several years while, during the second half of the war, at the Gottingen Aerodynamics Research Institute, I had to work on vibration problems. I am indebted to that time for valuable discoveries: on the one hand the applicability of abstract tools to the solution of concrete problems, on the other hand, the—for a pure mathematician—unexpected difficulty and un- accustomed responsibility of numerical evaluation. It was a matter of estimating eigenvalues of non-selfadjoint differential equations and matrices. I attacked the more general problem of developing a metric spectral theory, to begin with for finite complex matrices; few of the results have yet been published. A by-product was a topology-free spectral theory for a particular class of operators; this too is only partly published. I hope to be able to return to these topics when my algebraic work has reached a certain conclusion.
To finish this report it is perhaps appropriate to say a few words to place my work into its general context. The main line of development of mathematics has been characterized for several decades by the invasion of the axiomatic method into ever more areas. The goal is to derive all of mathematics deductively from just a few basic principles such as order and continuity. By turning increasingly towards the abstract, revolutionary unification has been achieved in mathematics, which must, by the way, gradually reach out also into school mathematics. It is as if some areas of mathematics which earlier could hardly be reached on foot are now connected by motorways. My own work has contributed nothing to these significant develop- ments, except perhaps the recently undertaken attempt to free the theory of permuta- tion groups from its restriction to finite groups. In fact, the impetus which Gottingen had given to abstract algebra reached Berlin just when I was a student, and recogni- tion of the implications of the axiomatic method fascinated me just as it did my fellow students. But I could not share the general opinion that this would henceforth be the only rewarding direction for research. It seemed to me that, like all great deductive systems, it was threatened by the danger that the problems which it could not prop- erly accommodate would be dismissed as uninteresting, whereas on the contrary, these ought to provide a stimulus to broaden the foundations. The development of the theory of finite groups has shown, I think, that this point of view was not wholly unjustified. The circle of questions in all its detail proved to be hardly accessible by the extensive apparatus that had been developed for the axiomatic method, and it failed to retain the interest of group-theorists, most of whom turned to infinite groups, in particular to continuous groups. It has, however, been revived by Hall and others using classical methods and posing classical questions, and the theory of finite groups, if isolated, has nevertheless undergone an almost dramatic development in recent years. In terms of the metaphor I used earlier, this research area seems to me to
Helmut Wielandt's Acceptance of Membership XV
be a mountainous region that is still undisturbed by roads and has to be traversed on foot. But this has its charm. And the nice surprises that one experiences compensate for the occasional compassionate glances of motorists.
This has been an admittedly only subjective justification of the direction of my work. But it is unmistakable that questions about finite structures are again coming strongly to the fore also in other areas of mathematics, partly as a result of growing applications of computing machines. I am convinced that the "finite" direction will be reunited with the mainstream in the course of the next few decades.
