# Daniel Gorenstein

### Quick Info

Boston, Massachusetts, USA

Martha's Vineyard, Massachusetts, USA

**Daniel Gorenstein**was an American mathematician best known for his part in the classification of finite simple groups.

### Biography

**Daniel Gorenstein**became interested in mathematics at a young age. He taught himself calculus at the age of 12 years. His secondary schooling was at Boston Latin School from which he graduated, entering Harvard University. There he worked under Saunders Mac Lane and became interested in finite groups, the subject he would return to after a few years studying algebraic geometry to make it his life's major work.

Gorenstein graduated in 1943 and, to contribute to the war effort, he accepted a teaching position at Harvard to teach mathematics to army personnel. At the end of World War II, Gorenstein returned to Harvard, this time to undertake graduate work with Zariski. This work led to a thesis on algebraic geometry in which he introduced rings which are now named after him.

After the award of his doctorate in 1950 Gorenstein accepted a post at Clark University in 1951. He remained there, except for the year 1958-59 which he spent as a visiting professor at Cornell, until he moved to Northeastern University in 1964. In 1968-69 he was a member of the Institute for Advanced Study at Princeton. After five years at Northeastern, Gorenstein accepted a professorship at Rutgers University where he remained until his death.

At Rutgers, Gorenstein was chairman of the Mathematics Department from 1975 until 1981. In 1984 Rutgers appointed him as Jacqueline B Lewis Professor of Mathematics and then, in 1989, he became Director of the newly founded National Science Foundation Science Technology Center in Discrete Mathematics and Theoretical Computer Science. This Center was a joint project between Rutgers University and Princeton University with AT&T Bell Laboratories and Bell Communications research as partners.

He returned from algebraic geometry to his early research topic of finite groups in 1957, stimulated by a collaboration with Herstein. His involvement in the classification of finite simple groups began in the year 1960-61 when he attended the

*Group Theory Year*at the University of Chicago. He wrote [3]:-

My first foray into simple group theory dated from the famous 1960-61 group theory year at the University of Chicago, during which Walter Feit and John Thompson settled the solvability of groups of odd order. It was there that I began a long collaboration with John and Walter and met many of the leaders in the field: Brauer, Suzuki, Graham Higman, and Ito. (It was only somewhat later that I met Philip Hall and Wielandt.) Alperin was spending the year in Chicago to write his thesis with Higman, while still a graduate student at Princeton.The classification of finite simple groups involved contributions by a host of mathematicians world wide. However it was Gorenstein who took an overview of the whole project and steered it to a successful conclusion. It is for the classification of finite simple groups that his name will always be remembered, certainly the mathematical achievement of the 20th century. If Gorenstein was the man with the best overview of this achievement, then surely we can do no better than to quote his own description of events. We quote from his response to the award of a Steele prize in 1989 given in [3]:-

Largely under the impetus of the odd order theorem, there was an awakening interest in finite group theory. Throughout the next decade and a half a long list of gifted young mathematicians, who were to play a prominent role in the classification proof, were attracted to the field. In the United States, John Thompson had a string of outstanding graduate students: Sims, Goldschmidt, Lyons, Griess. Glauberman was a student of Bruck's at the beginning of the period and Aschbacher near the end. Ronald Solomon wrote his thesis with Feit, Seitz with Curtis, Stephen Smith with Higman at Oxford, O'Nan with me, and Shult was essentially self-taught.Gorenstein's books on finite groups and the classification of finite simple groups are

But the attraction was not limited to the United States. Janko in Australia. Conway in England, and Fischer in Germany, each discovering three new sporadic groups, stimulated considerable additional interest, leading to an intensification of the search for further simple groups. Tits (entering the field somewhat earlier) had deepened our understanding of the Chevalley groups and their Steinberg-Suzuki-Ree variations, Bender in Germany was to prove the fundamental strongly embedded subgroup classification theorem, and Harada was beginning his career in Japan. At the end of the period, there were a number of others: Foote from Canada working with John Thompson in Cambridge, England, Geoffrey Mason from England, coming to the United States, and writing his thesis with Fong, himself a student of Brauer, and in Germany, Timmesfeld and Stellmacher, students of Fischer, and Stroth, a student of Huppert, but writing his thesis on a problem suggested by Held, who had himself been a student of Janko.

There were a great many other group theorists as well who made significant contributions to the classification proof. But it was Aschbacher's entry into the field in the early 1970s that irrevocably altered the simple group landscape. Quickly assuming a leadership role in a single minded pursuit of the full classification theorem, he was to carry the entire "team" along with him over the following decade until the proof was completed.

I was fortunate indeed to have interacted in one way or another during this twenty year period with most of the mathematicians I have mentioned.

Simultaneously with this burgeoning research effort, finite simple group theory was establishing a well-deserved reputation for inaccessibility because of the inordinate lengths of the papers pouring out. The 255 page proof of the odd order theorem, filling an entire issue of the Pacific Journal, had set the tone, but it was by far not the longest paper. Moreover, the techniques being developed, no matter how seemingly powerful for the problems at hand, appeared to have no applications outside finite group theory. Although there was admiration within the mathematical community for the achievements, there was also a growing feeling that finite group theorists were off on the wrong track. No mathematical theorem could require the number of pages these fellows were taking! Surely they were missing some geometric interpretation of the simple groups that would lead to a substantially shorter classification proof.

The view from the inside was quite different: all the moves we were making seemed to be forced. It was not perversity on our part, but the intrinsic nature of the problem that seemed to be controlling the directions of our efforts and shaping the techniques being developed.

*Finite groups*(1968),

*Finite simple groups : an introduction to their classification*(1982),

*The local structure of finite groups of characteristic 2 type*(jointly written with Richard Lyons) (1983) and

*The classification of the finite simple groups*(jointly written with Richard Lyons and Ronald Solomon) (1994).

Gorenstein received many honours for his work. In 1972-73 he was both Guggenheim Fellow and Fulbright Research Scholar. During 1978 he was Sherman Fairchild Distinguished Scholar at the California Institute of technology. He was elected to the National Academy of Sciences (1978) and the American Academy of Arts and Sciences (1978). He also received the Steele Prize from the American Mathematical Society in 1989. There are three Steele Prizes awarded and Gorenstein received the award for expository mathematical writing at the American Mathematical Society Summer Meeting in Boulder, Colorado, USA. The citation for the award states [3]:-

Gorenstein was a major figure in setting the direction of the classification program. He coordinated the activities in the program, functioning as the "coach" to a team, with optimism, perseverance, and technical power. His expository articles and books ... are beautiful accounts of this fantastic intellectual adventure. His presentation of theorems and definitions, as well as the flow of argument and the evolution of ideas, is precise and generous and reaches out to the reader.

### References (show)

- M Aschbacher, Daniel Gorenstein (1923-1992),
*Notices Amer. Math. Soc.***39**(10) (1992), 1190-1191. - R Lyons, Tribute to Daniel Gorenstein,
*The Gel'fand Mathematical Seminars, 1990-1992*(Boston, 1993), ix-x. - 1989 Steele prizes awarded at Summer Meeting in Boulder,
*Notices Amer. Math. Soc.***36**(7) (1989), 831- 836.

### Additional Resources (show)

Other websites about Daniel Gorenstein:

### Honours (show)

Honours awarded to Daniel Gorenstein

Written by J J O'Connor and E F Robertson

Last Update May 2000

Last Update May 2000