# Gorenstein, Daniel

# GORENSTEIN, DANIEL

(*b*. Boston, Massachusetts, 1 January 1923; *d*. Martha’s Vineyard, Massachusetts, 26 August 1992),

*mathematics, algebra, group theory*.

Gorenstein was an American mathematician whose main research was in the theory of finite groups, where he focused single-mindedly on the search for simple groups. Besides contributing major theorems himself, he mapped out the path that would lead to the goal of a complete classification, and became the informal leader of a disseminated team which achieved that goal.

**Private Life** . Born and bred in Boston, Daniel Gorenstein attended the Boston Latin School, from which he entered Harvard College, earning his AB in 1943 and PhD in 1950. His doctoral advisor was Oscar Zariski and his thesis was on a problem in algebraic geometry and its foundations in commutative algebra, a subject that he did not pursue after his first published paper. He married Helen Brav in 1947; they had four children, three girls and a boy. In his obituary of Gorenstein (*The Independent*, London 1992), Michael Collins wrote “He was a short stocky man, who applied the same muscular techniques to his mathematics as to his tennis,” and “Gorenstein both worked hard and played hard. […] Research was over by lunchtime. Living in Portland Place [London] in 1972–73 must have been ideal, for he could then devote the rest of the day to seminars and to visits to art galleries, museums, restaurants and the theatre. He collected modern art, with a good eye for artists yet to be recognised, and for the value of the works of those who were.”

**Career** . Soon after he achieved his doctorate Gorenstein became assistant professor at Clark University in Worcester, Massachusetts. He remained there until 1964, when he became professor at Northeastern University in Boston. In 1969 he moved to Rutgers University, New Brunswick, New Jersey, where he remained for the rest of his life, serving as chairman of the mathematics department from 1975 to 1981 and director of DIMACS, the NSF Center for Discrete Mathematics and Computer Science, from 1989 until his death. In addition, he held visiting positions in other parts of the world: Cornell University, 1958–1959; Princeton Institute for Advanced Study, 1968–1969; London and the Weizmann Institute, 1972–1973 as Guggenheim Fellow and Fulbright Research Scholar; and California Institute of Technology, 1978. He served at various times also as consultant to the MIT Lincoln Laboratory and to the Institute for Defense Analysis. His distinction was recognized by election to the National Academy of Sciences and to the American Academy of Arts and Sciences in 1987, by the award of the 1989 Steele Prize for Expository Writing by the American Mathematical Society, and by countless invitations to deliver lectures at prestigious national and international mathematical meetings.

**Mathematics** . For a mathematician Gorenstein was unusually slow to establish himself. He published one paper arising out of his doctoral research in algebraic geometry. Its influence is still visible in the term *Gorenstein ring*, but he left the area and moved on to coding theory and finite group theory. In the former he wrote two papers jointly with colleagues from the Lincoln Laboratory, one of which, on generalized BCH codes published in 1961, introduced a significant decoding algorithm and was reissued thirteen years later in a collection titled *Key Papers in the Development of Coding Theory*, edited by Elwyn Berlekamp. In finite group theory his first dozen papers contain respectable but generally inconsequential contributions. In his late thirties, however, inspired by what he heard and by the colleagues he met at the 1960–1961 group-theory year at the University of Chicago, his attention became focused on the search for finite simple groups. His first major contribution was joint work with John H. Walter of the University of Illinois, classifying the simple groups whose Sylow 2-subgroups are dihedral. Their four papers on the subject come to more than 200 pages in print, not as long as the paper by Walter Feit and John G. Thompson on groups of odd order, but nevertheless an indication of what was to come. These papers made a major impact, important not only for the theorems they contain but for the development of local methods, in particular the so-called signalizer-functor theory, that exploited ideas introduced by Philip Hall and Graham Higman in 1956 and by John G. Thompson in 1959 and 1962. Further huge projects followed: For example, with J. L. Alperin and Richard Brauer he classified the simple groups in which the Sylow 2-subgroups are quasi-dihedral or wreathed (261 pages); with Koichiro Harada he classified the simple groups in which every 2-subgroup can be generated by four elements (464 pages), a work which comprehensively dealt with all small Sylow 2-subgroups. His contributions in this area were many and varied, classifying simple groups either by the structure of their Sylow 2-subgroups or by the structure of the centralisers of their elements of order 2.

Significant though this research was, it was not his most important contribution. Although he had made himself a master, he was not uniquely qualified for the technical work; others could have done it. What made him unique were his vision and his leadership. Some time late in the 1960s he began to formulate a strategy for the classification of the finite simple groups. This was gradually refined and formulated, expounded in lectures in 1972 first in Chicago, then in London and at the Weizmann Institute in Israel, and published in a paper in the *Israel Journal of Mathematics* in 1974. There were many who doubted. But in the end his imagination proved correct. Although much of the technical work was done by many others, among whom Michael Aschbacher was foremost, it was Gorenstein (“The Godfather”) who had the personality and the ability to see the whole picture, to inspire and direct, to keep an army of research workers in many parts of the world working individually but together to ensure that none of the many cases was forgotten. The conclusion of the project was announced with a mixture of triumph and diffidence in 1980. Triumph because it did seem that methods adequate to solve the problem had been developed, that all the theorems had been proved, that no cases had been overlooked. Diffidence because all involved knew that some papers were yet to be published, and that the published papers, lengthy and technical as they were, were bound to contain mistakes. Indeed, as emerged later, one significant piece of the structure was missing and had to be supplied in a two-volume work on quasi-thin groups by Aschbacher and Stephen Smith.

At this point Gorenstein recognized that his task was only half done. He wrote books and papers surveying the work and explaining the classification. It was for this that he was awarded the Steele Prize for Mathematical Exposition. In his response to the Steele Prize citation he wrote:

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. […] 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!

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.

Thus in writing about the classification for the general mathematical audience, I had a dual motivation: on the one hand, to convey the nature of the solution as it was unfolding, both the methods involved and the striking results themselves, and, on the other, to attempt to convince the larger community that the internal inductive approach we were taking to the classification was, despite its resulting length, the only viable one for establishing the desired theorem. (American Mathematical Society, p. 833)

Just as important as his expository writing, however, was Gorenstein’s instigation of the project to produce a “new generation” proof of the classification. The idea was to produce a shorter, self-contained, properly structured proof, with all the gaps filled and all the slips and errors removed. He recruited Richard Lyons of Rutgers University and Ronald Solomon of Ohio State University and they began the project together. Much of the work was written at the time of his death but much was still to be done, and the first of a projected eleven volumes (six of which had been published by December 2006), did not appear until two years later. But the volumes carry his name as coauthor, a fitting tribute to his vision and his huge contribution to mathematics.

## BIBLIOGRAPHY

*Gorenstein’s writings are listed by* Mathematical Reviews. *In addition to his books, there are approximately seventy research articles, most of them on simple groups*.

### WORKS BY GORENSTEIN

*Finite Groups*, 2nd ed. New York: Chelsea Publishing, 1980. (Orig. New York: Harper & Row, 1968.)

As editor. *Reviews on Finite Groups: As Printed in Mathematical Reviews*, 1940 through 1970, Volumes 1–40 Inclusive. Providence, RI: American Mathematical Society, 1974.

*Finite Simple Groups: An Introduction to Their Classification*. New York: Plenum, 1982. Russian translation by V. I. Loginov. *Konechnye prostye gruppy*. Moscow: Mir, 1985.

*The Classification of Finite Simple Groups*. Vol. 1, *Groups of Noncharacteristic 2 Type*. New York: Plenum, 1983.

With Richard Lyons and Ronald Solomon. *The Classification of the Finite Simple* Groups, vols. 1–6. Providence, RI: American Mathematical Society, 1994–2006. (Eleven volumes are planned.)

### OTHER SOURCES

American Mathematical Society. “Citation for the 1989 Steele Prize for Expository Writing.” *Notices of the American Mathematical Society* 36 (1989): 831–833.

Aschbacher, Michael. “Daniel Gorenstein, 1932–1992.” *Notices of the American Mathematical Society* 39 (1992): 1190. Obituary.

Collins, Michael. “Professor Daniel Gorenstein.” *The Independent* (London), 9 September 1992. Obituary.

*Peter Neumann*

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**Gorenstein, Daniel**