John G. Thompson
Encyclopedia
John Griggs Thompson is a mathematician
at the University of Florida
noted for his work in the field of finite group
s. He was awarded the Fields Medal
in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize
.
from Yale University
in 1955 and his doctorate
from the University of Chicago
in 1959 under the supervision of Saunders Mac Lane
. After spending some time on the Mathematics faculty at the University of Chicago, he moved in 1970 to the Rouse Ball Professorship in Mathematics
at the University of Cambridge
, England
, and later moved to the Mathematics Department of the University of Florida
as a Graduate Research Professor. He is currently a Professor Emeritus of Pure Mathematics
at the University of Cambridge, and professor
of mathematics
at the University of Florida. He received the Abel Prize
2008 together with Jacques Tits
.
of Frobenius kernels
. At the time, this achievement was noted in The New York Times
(though his university affiliation was stated incorrectly there).
Thompson became a key figure in the progress toward the classification of finite simple groups
. In 1963, he and Walter Feit
proved that all nonabelian
finite simple group
s are of even order
(the Odd Order Paper
, filling a whole issue of the Pacific Journal of Mathematics
). This work was recognised by the award of the 1965 Cole Prize
in Algebra of the American Mathematical Society
. His monumental N-group papers
classified all finite simple groups for which the normalizer
of every non-identity solvable
subgroup
is solvable. This included, as a by-product, the classification of all minimal finite simple groups (those for which every proper subgroup is solvable). This work had great influence on later developments in the classification of finite simple groups, and was quoted in the citation by Richard Brauer
for the award of Thompson's Fields Medal in 1970 (Proceedings of the International Congress of Mathematicians
, Nice
, France
, 1970).
The Thompson group
Th is one of the 26 sporadic
finite simple groups. Thompson also made major contributions to the inverse Galois problem
. He found a criterion for a finite group to be a Galois group
, that in particular implies that the monster simple group
is a Galois group.
. In 1982, he was awarded the Senior Berwick Prize of the London Mathematical Society
, and in 1988, he received the honorary degree of Doctor of Science
from the University of Oxford
. Thompson was awarded the United States National Medal of Science
in 2000. He is a Fellow of the Royal Society (United Kingdom
), and a recipient of its Sylvester Medal
. He is a member of the Norwegian Academy of Science and Letters
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
at the University of Florida
University of Florida
The University of Florida is an American public land-grant, sea-grant, and space-grant research university located on a campus in Gainesville, Florida. The university traces its historical origins to 1853, and has operated continuously on its present Gainesville campus since September 1906...
noted for his work in the field of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s. He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
.
Biography
He received his B.A.Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
from Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
in 1955 and his doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
from the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
in 1959 under the supervision of Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
. After spending some time on the Mathematics faculty at the University of Chicago, he moved in 1970 to the Rouse Ball Professorship in Mathematics
Rouse Ball Professor of Mathematics
The Rouse Ball Professorship of Mathematics is one of the senior chairs in the Mathematics Departments at the University of Cambridge and the University of Oxford. The two positions were founded in 1927 by a bequest from the mathematician W. W. Rouse Ball...
at the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
, and later moved to the Mathematics Department of the University of Florida
University of Florida
The University of Florida is an American public land-grant, sea-grant, and space-grant research university located on a campus in Gainesville, Florida. The university traces its historical origins to 1853, and has operated continuously on its present Gainesville campus since September 1906...
as a Graduate Research Professor. He is currently a Professor Emeritus of Pure Mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
at the University of Cambridge, and professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University of Florida. He received the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
2008 together with Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
.
Work
Thompson's doctoral thesis introduced powerful and innovative techniques, and included the solution of a problem in finite group theory which had stood for around sixty years, the nilpotencyNilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
of Frobenius kernels
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
. At the time, this achievement was noted in The New York Times
The New York Times
The New York Times is an American daily newspaper founded and continuously published in New York City since 1851. The New York Times has won 106 Pulitzer Prizes, the most of any news organization...
(though his university affiliation was stated incorrectly there).
Thompson became a key figure in the progress toward the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. In 1963, he and Walter Feit
Walter Feit
Walter Feit was a Jewish Austrian-American mathematician who worked in finite group theory and representation theory....
proved that all nonabelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s are of even order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
(the Odd Order Paper
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
, filling a whole issue of the Pacific Journal of Mathematics
Pacific Journal of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by a number of American, Asian and Australian universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation.It was...
). This work was recognised by the award of the 1965 Cole Prize
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...
in Algebra of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
. His monumental N-group papers
N-group (finite group theory)
In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups...
classified all finite simple groups for which the normalizer
Centralizer and normalizer
In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively...
of every non-identity solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
is solvable. This included, as a by-product, the classification of all minimal finite simple groups (those for which every proper subgroup is solvable). This work had great influence on later developments in the classification of finite simple groups, and was quoted in the citation by Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
for the award of Thompson's Fields Medal in 1970 (Proceedings of the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
, Nice
Nice
Nice is the fifth most populous city in France, after Paris, Marseille, Lyon and Toulouse, with a population of 348,721 within its administrative limits on a land area of . The urban area of Nice extends beyond the administrative city limits with a population of more than 955,000 on an area of...
, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
, 1970).
The Thompson group
Thompson group (finite)
In the mathematical field of group theory, the Thompson group Th, found by and constructed by , is a sporadic simple group of orderThe centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson...
Th is one of the 26 sporadic
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
finite simple groups. Thompson also made major contributions to the inverse Galois problem
Inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century, is unsolved....
. He found a criterion for a finite group to be a Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
, that in particular implies that the monster simple group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
is a Galois group.
Further recognition
In 1971, Thompson was elected to the United States National Academy of SciencesUnited States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
. In 1982, he was awarded the Senior Berwick Prize of the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
, and in 1988, he received the honorary degree of Doctor of Science
Doctor of Science
Doctor of Science , usually abbreviated Sc.D., D.Sc., S.D. or Dr.Sc., is an academic research degree awarded in a number of countries throughout the world. In some countries Doctor of Science is the name used for the standard doctorate in the sciences, elsewhere the Sc.D...
from the University of Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...
. Thompson was awarded the United States National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
in 2000. He is a Fellow of the Royal Society (United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
), and a recipient of its Sylvester Medal
Sylvester Medal
The Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize...
. He is a member of the Norwegian Academy of Science and Letters
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
.
See also
- Feit–Thompson theoremFeit–Thompson theoremIn mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
- McKay–Thompson series
- Quadratic pairQuadratic pairIn mathematical finite group theory, a quadratic pair for the odd prime p, introduced by , is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimum polynomial 2...
- Thompson subgroupThompson subgroupIn mathematical finite group theory, the Thompson subgroup J of a finite p-group P refers to one of several normal subgroups. originally defined of J to be the subgroup generated by the abelian subgroups of P of maximal order...
- Thompson order formulaThompson order formulaIn mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson, gives a formula for the order of a finite group in terms of the centralizers of involutions.-Statement:...
- Thompson uniqueness theoremThompson uniqueness theoremIn mathematical finite group theory, Thompson's original uniqueness theorem states that in a simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3....