**Hans Zassenhaus**'s parents were Julius Zassenhaus and Margarete E F Ziegler. Julius Zassenhaus was a historian who was the principal of a girls' gymnasium. He wrote an anthology on the history of Christianity and also had taught and written about Albert Schweitzer's "reverence for life". He wrote the 3-part work, *Evangelisches Religionsbuch für höhere Schulen* Ⓣ published in 1929-30. Julius and Margarete had four children, sons Hans, Guenther and Willfried, and the youngest member of the family was a daughter Hiltgunt born in 1916. Hiltgunt wrote of her mother [3]:-

My mother was a very special person, I don't think anyone who knew her will ever forget her. You may think this is the blind adoration of a daughter, but it is not. She helped to shape the lives of many people by her example and her kindness.

Hans spent the first four years of his life in Koblenz-Moselweiss before the family moved to Hamburg in 1916. His secondary education was at two schools in Hamburg, graduating from the Lichtwarkschule in 1930. The Lichtwarkschule (founded in 1914 and named after Alfred Lichtwark) was a reformist educational school in the Hamburg-Winterhude district. The headmaster was Heinrich Landahl from 1926. After graduating from this school, Hans entered the University of Hamburg in the autumn of 1930. At first he studied mathematics and physics with the intention of specialising in atomic physics. However he had fine mathematics teachers in Emil Artin and Erich Hecke and, particularly Artin, inspired him to undertake research in mathematics. Zassenhaus studied for his doctorate under Artin's supervision. During this time he proved the Zassenhaus (butterfly) lemma, a beautiful result on subgroups which can be used to give a simple, and very beautiful, proof of the Jordan-Hölder theorem. He published this in the 3-page paper *Zum Satz von Jordan-Hölder-Schreier* Ⓣ (1934). However, these years were extremely difficult ones for the Zassenhaus family for two reasons.

The first problem was that Adolf Hitler came to power in 1933. Hiltgunt described the Zassenhaus family's first encounter with the Nazis [3]:-

One morning in winter I awoke, and I could see no sky through my window. The room had a dim, yellowish, almost sulfur-like light. The window glass was covered with thick, yellow paper, printed over and over with swastikas. Overnight our house had been dimmed by unknown Nazi hands. This was in Hamburg, Germany,30January1933. Adolf Hitler had come to power. Two SS men came the same day and asked for my father. As he was sick at that time, they let him stay ... That day he said, "From now on our house will be a fortress, where we will live and think as we did before." My mother added, "It will last only a few months." It lasted twelve years.

The second problem for the Zassenhaus family was the deterioration in Julius Zassenhaus's health. Julius had contracted influenza in the epidemic of 1918 and, although he had recovered, this had undermined his health. As the above quote indicates, he was sick in January 1933 and his doctor, Dr Stromberger, diagnosed the slowly worsening illness to be Parkinson's Disease. Soon, he had to resign his position as school principal. By 1933 all four Zassenhaus children were at Hamburg University, Hans was working on his doctoral dissertation, Guenther and Willfried were studying medicine and Hiltgunt began studying Scandinavian languages. Hiltgunt wrote [3]:-

In Germany a new law had been passed: only students who participated in Nazi activities would get scholarships. That left us out. We began to wonder how we would manage to pay our tuition. Together we made an all-out effort. Expenses were cut to a minimum. There was meat only once a week, and when my mother had divided it into six equal portions, the plate seemed rather empty, but less expensive food would do. We moved into a much smaller house[in the Lyserstrasse]. Fortunately, it still had a room for each of us, even if there was a space for only a bed and a chair and a table.

The Nazis burned books in bonfires throughout the city, books such as those Julius Zassenhaus had written [3]:-

... containing the very ideas and thoughts which had governed our lives until then. More than the flames I remembered the fanatic stare in the eyes of the men who had burned them.

Despite all the turmoil in his life, Hans Zassenhaus completed his doctoral dissertation *Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen* Ⓣ on 28 July 1934. In it he considered permutation groups whose elements are determined by the images of three points. These groups are called Zassenhaus groups today. In his dissertation Zassenhaus classified all 3-fold transitive Zassenhaus groups. These groups play an important role in the classification of finite simple groups coordinated by Daniel Gorenstein.

From 1934 to 1936 Zassenhaus worked at the University of Rostock and wrote his famous group theory text *Lehrbuch der Gruppentheorie* Ⓣ (1937) based on Artin's lectures at Hamburg. Philip Hall reviewed the book [7]:-

Anyone opening this slender book expecting merely to find an account of the more elementary and standard parts of the subject will be speedily disillusioned. Although the volume before us is only the first part of an extended treatise, it already takes the reader from the foundations right through to some of the most important advances of the last few years, results which now find their place in a textbook for the first time. To have compressed so much material, both new and old, into so small a space is indeed a remarkable feat. And it may be that the economy of the style will make some parts of the book not altogether easy reading. But probably the main credit for this high " specific content " is due to the systematic grouping of the exposition around the central notion of a homomorphic correspondence. This is an arrangement which not only lends the book a close-knit logical structure: it also serves to correct the impression which group theory sometimes gives of being a sequence of beautiful theorems having little to do with one another. Such an impression could scarcely survive a reading of this volume. Both as regards originality of treatment and thorough modernity of outlook, this promises to be when complete a contribution of first-rate importance to the literature of the subject, and one which all serious students of groups will wish to possess. ... Herr Zassenhaus has rendered a most valuable service to all who are interested in this subject, and we shall await the publication of the later volumes with impatience.

Zassenhaus became Artin's assistant at Hamburg in 1936. His habilitation thesis of 1938 studied Lie rings of prime characteristic. It was published as *Über Liesche Ringe mit Primzahlcharakteristik* Ⓣ (1939). Zassenhaus found that a normal academic career was made impossible for him because of his intense dislike of the Nazi party. We give an example of how he behaved during the war. Horst Tietz was expelled from Hamburg University by the Nazis. He asked Erich Hecke what he should do [1]:-

Hecke said that naturally Teitz should come secretly to his mathematical lectures, also those of Hans Zassenhaus ... Teitz was somewhat afraid of Zassenhaus, who always wore the insignia of a Nazi organisation, but Hecke calmed him by saying that Zassenhaus was their trusted agent who behaved in a politically correct manner so as to protect them. This in fact happened in1942, when somehow Tietz's presence as a secret student at the university became known and a denunciation was threatened: Zassenhaus warned Tietz, and Hecke immediately cancelled the class, giving back the student fees. In fact, Tietz only discovered much later that Zassenhaus was connected with a resistance effort that helped hide endangered people.

Hiltgunt wrote about these years [3]:-

We had learned to fear the ringing of the doorbell after dark. Friends had disappeared, picked up in their homes by the Gestapo in the middle of the night. They had not returned, but we had seen the letters from the Gestapo - form letters, informing the families that death had ensued due to a "sudden illness." We knew better and - though later it would be denied - many knew. There was a concentration camp on the outskirts of Hamburg. High walls topped with barbed wire surrounded it, but the screams could still be heard. ... One day, quite by accident, I learned that my mother was helping Jewish friends. I'd noticed that at times she went away for days, and that she sometimes looked very pale and exceedingly tired. My father had told us that we must help mother as much as we could, but I did not understand why.

The teaching Zassenhaus did at Hamburg University after 1940 was unofficial, he had to join the Nazi party if he was to keep his position and Zassenhaus was not prepared to do this. He resigned and joined the navy, working on weather forecasting during the remainder of World War II. He married to Lieselotte Lohmann (28 May 1914- 20 March 2006) in 1942; they had three children, Michael, Angela and Peter. When he was offered the chair of mathematics at Bonn in 1943 he requested that he be allowed to postpone a decision until the end of the war. After the war he did not accept the Bonn post, preferring that it went to someone who had lost their chair under the Nazis. After the war, he became chairman of the department at Hamburg, spending session 1948-49 at Glasgow in Scotland. Then, in 1949, he accepted a chair of mathematics in Montreal, Canada. Although he visited many other universities during the next 10 years, he remained in the post at Montreal until he moved to the University of Notre Dame in 1959 where he was appointed Peter Redpath Professor. He also served as director of the university Computing Center. He was Mershon Visiting Professor at The Ohio State University during the academic year 1963-64, then, at the end of his visit, he accepted the post of research professor at Ohio State University and held this position until he retired. During these years he held a number of visiting positions: Gauss Professor at Göttingen University, visiting professor at Heidelberg University, visiting professor at the University of California at Los Angeles, and visiting professor at Warwick University (where I [EFR] was fortunate enough to meet him for the first time).

We have mentioned his work in group theory and Lie rings above. The work on Lie rings extended to Lie algebras and he developed computational methods for studying them. In a long series of papers he applied Lie algebras to problems of theoretical physics. He wrote up his lectures on Lie theory as *Lie groups, Lie algebras and representation theory* (1981). Here are the chapter headings:

1. Survey of the classical theory of Lie groups and Lie algebras of finite dimension over the real number field.

2. Continuous groups.

3. General representation theory.

4. Representation theory of Lie algebras of zero characteristic.

5. Reflection groups and the classification of semisimple Lie algebras of zero characteristic.

His work on computational algebraic number theory seems to have started when he visited Caltec in 1959 and collaborated with Olga Taussky-Todd. They published joint papers such as *On the semigroup of ideal classes in an order of an algebraic number field* (1961), *On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field* (1962) and *On the 1-cohomology of the general and special linear groups* (1970). He put forward a programme to develop methods for computational number theory which, given an algebraic number field, involved calculating its Galois group, an integral basis, the unit group and the class group. He contributed himself in a major way to all four of these tasks. In collaboration with Michael Pohst, he published a number of papers on computational number theory and also the book *Algorithmic algebraic number theory* (1989). The authors write in the Preface:-

This book is a step in a new direction: to modify existing theory from a constructive point of view and to stimulate readers to make their own computational experiments. We are thoroughly convinced that their observations will help to build a new basis from which to venture into new theory on algebraic numbers. History shows that in the long run, number theory always followed a cyclic movement from theory to construction to experiment to conjecture to theory. Consequently, this book is addressed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to(constructive)algebraic number theory and is therefore especially suited as a textbook for a course on that subject. On the other hand, many parts go beyond an introduction and make the user familiar with recent research in the field.

Charles Parry writes in a review:-

Few, including experts in the subject, will find this book light reading. However, with increasing emphasis on computational aspects of number theory, it fills a void in the literature.

Zassenhaus worked on a broad range of topics and, in addition to those mentioned above, he worked on nearfields, the theory of orders, representation theory, the geometry of numbers and the history of mathematics. He loved teaching and wrote several articles on the topic such as *The concept of depth in teacher training* (1963) and *On the teaching of algebra at the pre-college level* (1975). The authors of [12] write:-

Under his leadership, 'The Journal of Number Theory' was established at Ohio State in1969. Professor Zassenhaus took a profound interest in teaching of mathematics at all levels. He directed a large number of Ph.D. dissertations, twenty at Ohio State. He was very kind to his students and was most generous in sharing his time and creative ideas. Each summer, for several years, he taught a selected group of gifted high school students.

In 1982, Zassenhaus retired but continued to be associated with Ohio State University. He continued to direct the thesis studies of students and remained an active researcher, publishing around 40 publications in his retirement. The last time I [EFR] met Zassenhaus was at the 'Computational group theory' Special Section of the American Mathematical Society Meeting held in Phoenix, United States, in January 1989. This was also the last time I heard him lecture.

He received many honours for his outstanding contributions. He received the Lester Ford Prize from the Mathematical Association of America and was elected a fellow of the Royal Society of Canada. He was a Distinguished Fairchild Scholar at the California Institute of Technology (1974-75), received the Distinguished American Scientist Award from the Alexander von Humboldt Foundation, was named Sesquicentennial University Professor at Montreal University and received the Distinguished Research Award from The Ohio State University. He was awarded honorary degrees by Ottawa University, McGill University, University of Saarbrucken and Rostock University.

Zassenhaus, who was a member of the First English Lutheran Church in Columbus, died in the Hospice of Columbus. His funeral service was held in the Rutherford-Corbin Chapel in Worthington.

**Article by:** *J J O'Connor* and *E F Robertson*

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