# Feit, Walter

# FEIT, WALTER

(*b*. Vienna, Austria, 26 October 1930; *d*. New Haven, Connecticut, 29 July 2004),

*mathematics, algebra, group theory*.

Feit was an American mathematician whose main research was in the theory of finite groups and, in particular, their representations and characters. He made major contributions to the search for finite simple groups, to the theory and applications of exceptional characters, and to applications of finite group theory, particularly in Galois Theory. His greatest contribution was joint work with John Thompson proving Burnside’s Conjecture that groups of odd order are soluble.

**Origins and Education** . Walter Feit’s parents were Paul and Esther Feit, shopkeepers in the Molkereistrasse in Vienna. After the Anschluss in March 1938 Austrian Jews were as much at risk as those in Germany, and Walter’s aunt, Pauline, who lived in Miami, Florida, kept urging them to join her in the United States. The risk grew after the Krystallnacht pogrom of 9 and 10 November 1938, but their shop had survived, and they stayed. Finally they sent Walter to England on the last of the Kindertransport trains from Vienna late in August or early in September 1939. They died in a concentration camp some time during the war.

After a brief stay with another aunt, Frieda, in London, Walter settled in a refugee hostel in Oxford. In 1943 he won a scholarship to the Junior Day Department of the Oxford Technical College, a school that had been founded in 1934 in the former buildings of the City of Oxford Technical School in St. Ebbe's. This had two hundred pupils by 1940, and applications outnumbered places by five to one. After the 1944 Education Act it became a Secondary Technical School and in 1954 it moved to larger premises adjacent to the Technical College (later Oxford Brookes University) in Headington and was renamed Cheney School. He recorded his teachers as having been very encouraging, and that it was here that he became passionately interested in mathematics.

Soon after his sixteenth birthday he migrated to the United States, where he had a great many relatives, arriving in New York on 29 December 1946. He moved in with his aunt Pauline and her family in Miami and, after graduating from high school there, entered the University of Chicago in September 1947, graduating with a bachelor’s and a master’s degree in mathematics in June 1951. From there he moved to the University of Michigan to study for a doctorate under the supervision of Richard Brauer. After one year Brauer moved to Harvard, and Feit’s formal supervision was transferred to Robert M. Thrall. He continued to work under Brauer’s direction, however, and presented his thesis in the summer of 1954. He married Sidnie Dresher on his twenty-seventh birthday soon after his army service ended. They had a daughter Alexandra, an artist, and a son Paul, a mathematician.

**Career** . Already in 1953, a year before his Michigan doctorate was complete, Feit was appointed to an instructor-ship at Cornell University. Apart from drafted service in the army 1955–1956, he remained there until 1964, when he moved to Yale University, where he stayed until his retirement in 2003. At Yale he served at various times as director of undergraduate studies, director of graduate studies, and chairman of the mathematics department. His distinction was recognized by the award of the 1965 American Mathematical Society Cole Prize in Algebra, awarded to Feit and John G. Thompson jointly for their 1963 paper “Solvability of groups of odd order”; by election to the National Academy of Sciences in 1977; and to the American Academy of Arts and Sciences. He traveled all over the world attending conferences, lecturing, and serving on both national and international committees. He served as editor of many journals, and, in particular, succeeded the founding editor, Graham Higman, as editor-in-chief of the *Journal of Algebra* from 1985 to 2000.

**Mathematics** . Feit’s early reputation was based on his work in character theory of finite groups, a subject which had been invented by Georg Frobenius in 1896, and developed by Frobenius and his student Issai Schur in Germany, by William Burnside in England, and later by Richard Brauer. Of considerable interest at the time were so-called Frobenius groups. A subgroup of a group is known as a TI subgroup if it intersects each of its conjugates (other than itself) trivially. A Frobenius group is a group with a TI subgroup that is its own normaliser, and Frobenius had shown that such a subgroup (later known as a Frobenius complement) has a normal complement (now known as the Frobenius kernel). His proof was based on the discovery that some of the characters of the group are closely related to the characters of the subgroup. In 1955, Michio Suzuki, building on work of Brauer, had generalized this work of Frobenius to define so-called “exceptional characters” of a group with a TI subgroup, and had found conditions under which restriction of characters produces an isometry to a set of characters of the subgroup. Feit found that this phenomenon could be seen also in groups that have suitably embedded Frobenius subgroups. He developed the theory of exceptional characters and combined it with the theorem proved in 1959 by John Thompson, that the Frobenius kernel is always nilpotent, to make progress with the problem of finding all the doubly transitive groups in which the stabiliser of three points is trivial. The one-point stabilisers in such groups are themselves Frobenius groups, and Feit used delicate character-theoretic arguments to prove that if the degree of the permutation group was odd then it had to be one more than a power of two. This was soon used by Suzuki to discover new groups of this kind and to complete the characterization of such groups.

At this time, the late 1950s and early 1960s, finite group theory was developing in many exciting directions. A number of significant problems were ripe for solution, and in many cases exceptional character theory was one of the main tools for the job. Feit’s expertise in this area led to a number of successful collaborations, for example with Marshall Hall and John Thompson in the characterization of groups in which the centraliser of every non-identity element is nilpotent, and with John Thompson in the characterization of finite groups having a self-centralising element of order three. What brought lasting fame far beyond the group-theoretic community, however, was the collaboration with John Thompson on groups of odd order. The proof of Burnside’s conjecture that there are no simple groups of odd composite order was, and remained as of 2007, a monumental work. It was the first of the really long and intricate proofs in that area, 255 printed pages of concentrated argument. The theorem itself is of the highest importance not merely as an elegant and decisive statement about groups of odd order, but also as providing the key to the classification of the finite simple groups of composite order—all such groups must contain elements of order two, and study of their centralisers and of the normalisers of two-subgroups provides the way in. Feit’s main contribution was the character theory, where he had not only to adapt known methods of exceptional character theory to the special situations that arose in the study, but also extend the general theory by the introduction of new isometries between certain sets of characters of a group and appropriate sets of characters of subgroups.

Although character theory and representation theory remained the basis of most of Feit’s later work, he contributed over a wide range. He wrote on combinatorics, classifying (in joint work with Graham Higman) certain generalized polygons important in the classification of BN-pairs of small rank, and contributing to the study of balanced incomplete block designs and codes. He wrote on diophantine equations and problems of elementary number theory. He wrote extensively on the inverse Galois problem, showing that many insoluble groups could be realized as Galois groups of polynomials over the rational number field. He used his extensive knowledge of finite groups to contribute new information on the well-known problem of universal algebra that asks which lattices can occur as congruence lattices of finite algebraic systems. But although he was a mathematician who contributed throughout algebra and its applications, it is the Odd Order Theorem which will ensure that his name remains known to later generations.

## BIBLIOGRAPHY

*Feit wrote approximately eighty research papers in mathematics. They are listed in* Mathematical Reviews. *The Feit papers are in the Archives of American Mathematics, Research and Collections division of the Center for American History, University of Texas at Austin, donated by Sidnie Feit*.

### WORKS BY FEIT

*Characters of Finite Groups*. New York: W. A. Benjamin, 1967.

*The Representation Theory of Finite Groups*. Amsterdam and New York: North-Holland, 1982. Translated into Russian, 1990.

### OTHER SOURCES

Scott, Leonard, Ronald Solomon, John Thompson, et al. “Walter Feit (1930–2004).” *Notices of the American Mathematical Society* 52 (2005): 728–735.

Walter Feit Biographical Information. Available at http://www.math.yale.edu/public_html/WalterFeit/WalterFeit.html.

*Peter Neumann*

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