William Burnside
Encyclopedia
William Burnside was an English
mathematician
. He is known mostly as an early contributor to the theory of finite group
s.
Burnside was born in London
, and attended St. John's and Pembroke College
s at the University of Cambridge
, where he was Second Wrangler in 1875. He lectured at Cambridge for the following 10 years, before being appointed professor of mathematics at the Royal Naval College
in Greenwich
. While this was a little outside the main centers of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career.
Burnside's early work was in applied mathematics
. This work was of sufficient distinction to merit his election as a fellow of the Royal Society
in 1893, though it is little remembered today. Around the same time as his election his interests turned to the study of finite group
s. This was not a widely studied subject in late 19th century Great Britain
, and it took some years for his work in this area to gain widespread recognition.
The central part of Burnside's group theory work was in the area of group representation
s, where he helped to develop some of the foundational theory, complementing and sometimes competing with the work of Frobenius
, who began the subject in the 1890s. One of his best known contributions to group theory is his paqb theorem (which shows that every finite group whose order is divisible by fewer than three distinct primes is solvable
).
In 1897 Burnside's classic work Theory of Groups of Finite Order was published. The second edition (published in 1911) was for many decades the standard work in the field. A major difference between the editions was the inclusion of character theory
in the second.
Burnside is also remembered for the formulation of Burnside's problem
(which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it) and for Burnside's lemma
(a formula relating the number of orbits of a permutation group
acting on a set with the number of fixed points of each of its elements) though the latter had been discovered earlier and independently by Frobenius and Cauchy
.
In addition to his mathematical work, Burnside was a noted rower; while he was a lecturer at Cambridge he also coached the crew team. In fact, his obituary in The Times
took more interest in his athletic career, calling him "one of the best known Cambridge athletes of his day".
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...
mathematician
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....
. He is known mostly as an early contributor to the theory 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.
Burnside was born in London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, and attended St. John's and Pembroke College
Pembroke College, Cambridge
Pembroke College is a constituent college of the University of Cambridge, England.The college has over seven hundred students and fellows, and is the third oldest college of the university. Physically, it is one of the university's larger colleges, with buildings from almost every century since its...
s 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...
, where he was Second Wrangler in 1875. He lectured at Cambridge for the following 10 years, before being appointed professor of mathematics at the Royal Naval College
Old Royal Naval College
The Old Royal Naval College is the architectural centrepiece of Maritime Greenwich, a World Heritage Site in Greenwich, London, described by the United Nations Educational, Scientific and Cultural Organisation as being of “outstanding universal value” and reckoned to be the “finest and most...
in Greenwich
Greenwich
Greenwich is a district of south London, England, located in the London Borough of Greenwich.Greenwich is best known for its maritime history and for giving its name to the Greenwich Meridian and Greenwich Mean Time...
. While this was a little outside the main centers of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career.
Burnside's early work was in applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
. This work was of sufficient distinction to merit his election as a fellow of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
in 1893, though it is little remembered today. Around the same time as his election his interests turned to the study 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. This was not a widely studied subject in late 19th century Great Britain
Great Britain
Great Britain or Britain is an island situated to the northwest of Continental Europe. It is the ninth largest island in the world, and the largest European island, as well as the largest of the British Isles...
, and it took some years for his work in this area to gain widespread recognition.
The central part of Burnside's group theory work was in the area of group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s, where he helped to develop some of the foundational theory, complementing and sometimes competing with the work of Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, who began the subject in the 1890s. One of his best known contributions to group theory is his paqb theorem (which shows that every finite group whose order is divisible by fewer than three distinct primes is 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...
).
In 1897 Burnside's classic work Theory of Groups of Finite Order was published. The second edition (published in 1911) was for many decades the standard work in the field. A major difference between the editions was the inclusion of character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
in the second.
Burnside is also remembered for the formulation of Burnside's problem
Burnside's problem
The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group...
(which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it) and for Burnside's lemma
Burnside's lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
(a formula relating the number of orbits of a permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
acting on a set with the number of fixed points of each of its elements) though the latter had been discovered earlier and independently by Frobenius and Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
.
In addition to his mathematical work, Burnside was a noted rower; while he was a lecturer at Cambridge he also coached the crew team. In fact, his obituary in The Times
The Times
The Times is a British daily national newspaper, first published in London in 1785 under the title The Daily Universal Register . The Times and its sister paper The Sunday Times are published by Times Newspapers Limited, a subsidiary since 1981 of News International...
took more interest in his athletic career, calling him "one of the best known Cambridge athletes of his day".