**William 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.

- William Burnside Senior was a merchant who lived at 7 Howley Place, Paddington, where William, the elder of his parents two sons, was born.
- Christ's Hospital was a school which took in boys whose parents were unable to pay the fees for a boarding school, so it was particularly appropriate for an orphan like Burnside.
- In 1873 he moved from St John's College to Pembroke College, not for academic reasons but rather because St John's had such an excellent rowing team that Burnside was not good enough to make their first boat.
- Burnside was, however, considered to have the most elegant mathematical style.
- They were to influence greatly the direction that Burnside's research was to take.
- Pembroke had an applied mathematics tradition, so a decision taken because of rowing was largely responsible for the direction that Burnside took in his mathematical teaching and research.
- 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.
- We mentioned that Burnside was an excellent oarsman, a '7' who captained Pembroke, but he was considered too light to make the University Boat so never earned a rowing blue.
- 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.
- He turned down two offers from Pembroke for after Stokes died in 1903, the College invited Burnside to take up the post of Master of the College.
- Burnside was elected a Fellow of the Royal Society in 1893 for his work on hydrodynamics and complex function 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.
- Let us now examine some more of Burnside's contributions to group theory.
- Burnside quickly recognised the importance of Frobenius's methods and he began to use character theory.
- 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.
- A 1994 Fields Medallist 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 which contains 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.
- Burnside did get back to his mathematics, publishing On a group of order 25920 and the projective transformations of a cubic surface later that year, but he died in 1927.
- Hall was to prove a very worthy successor to Burnside as the promoter of group theory in England.

Born 2 July 1852, Paddington, London, England. Died 21 August 1927, Cotleigh, West Wickham, Kent, England.

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Algebra, Group Theory, Origin England, Puzzles And Problems

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive