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Burnside wrote the first treatise on groups in English and was the first to develop the theory of groups from a modern abstract point of view.Burnside was of Scottish ancestry on his father's side. He graduated from Pembroke College Cambridge in 1875 as second wrangler, bracketed with George Chrystal. Burnside was considered to have the most elegant mathematical style. He was then awarded a fellowship at Pembroke which he held from 1875 to 1886.
Among his teachers at Cambridge were Stokes, Adams and Maxwell in Applied Mathematics and Cayley in Pure Mathematics. They were to influence greatly the direction that Burnside's research was to take.
His first paper in 1883 considered elliptic functions but after 1885, the year he was appointed professor of mathematics at the Royal Naval College at Greenwich, his work turned to hydrodynamics.
Much of Burnside's work on hydrodynamics involved the use of complex variable and in papers of 1891 and 1892 he considered the group of linear fractional transformations of a complex variable. His work quickly turned to the study of groups and from 1894 onwards he was to be occupied almost entirely with the study of group theory.
Burnside was elected a Fellow of the Royal Society in 1893 for his work on hydrodynamics and complex function theory. However his work on group theory quickly progressed and in 1897 he published The Theory of Groups of Finite Order , the first treatise on group theory in English. This work was to have a major influence in the development of group theory.
In 1899 Burnside was elected to the Council of the London Mathematical Society and in the same year the Society awarded him the De Morgan medal. He was to be President of the Society from 1904 to 1906 and continued to serve on the Council until 1917.
Frobenius started his development of the representation theory of groups and character theory in 1896. Burnside quickly recognised the importance of Frobenius's methods and he began to use character theory. One of his most important results, namely that groups of order pq are soluble, appeared in 1904. Special cases of this result had been proved by Sylow (the case n = 0 in 1872), Frobenius (the case n = 1 in 1895) and Jordan (the case n = 2 in 1898). Burnside conjectured that every finite group of odd order is soluble and it is not surprising that he failed to prove this result as it was not proved until 1962 when W. Feit and J.C. Thompson proved the result in a 300 page paper.
Much of group theory today still moves in directions set by Burnside. His famous 'Burnside Problem' on the finiteness of groups when the elements have fixed finite orders is still a major area of group theory research today. In fact a 1994 Fields Medalist E. Zelmanov was awarded his medal for work related to the Burnside problem.
If the first edition of The Theory of Groups of Finite Order was important, the second edition published in 1911 and containing a systematic development of the subject including Frobenius's character theory and Burnside's work using these methods, was a classic which is still widely read today.
During his life Burnside was to publish around 150 papers of which about 50 were on group theory. In fact in the latter years of his life he turned to probability theory and his first paper on the subject appeared in 1918. He left a complete manuscript of a book on probability which was published as The theory of Probability in the year after his death.
As an undergraduate Burnside was an excellent oarsman, a '7', but was considered too light to make the University Boat. He retained an interest in rowing and later in fishing. He always had a love of Scotland and continued to take fishing holidays there throughout his life. He married soon after he took up the Chair at Greenwich and he was survived by two sons and three daughters. Looking at Burnside's career perhaps the greatest surprise is that he turned down an offer from Pembroke to return to his old College, preferring to remain at Greenwich.
References (6 books/articles)
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Tell me about Burnside's part in the development of group theory
William Burnside was elected to the Royal Society of London in 1893. You can see a history of the Royal Society and a list of the members among the mathematicians in our archive.
He was awarded the Royal Medal of the Royal Society in 1904. You can see a history of the Royal Medal and a list of the winners in our archive.
W Burnside was the London Mathematical Society President in 1906  1908. You can see a history of the LMS and a list of the presidents.
He was the winner of the London Mathematical Society De Morgan Medal in 1899. You can see a history of the LMS De Morgan Medal and a list of the winners.