VI.60 William Burnside
b. London, 1852; d. West Wickham, England, 1927
Theory of groups; character theory; representation theory
Burnside’s mathematical abilities first showed themselves at school. From there he won a place at Cambridge, where he read for the Mathematical Tripos and graduated as 2nd Wrangler in 1875. For ten years he remained in Cambridge as a Fellow of Pembroke College, coaching student rowers and mathematicians. In 1885, having published three very short papers, he was appointed professor at the Royal Naval College, Greenwich. He married in 1886 and the next year, at the age of thirty-five, he embarked on his career as a productive mathematician. He was elected as a Fellow of the Royal Society in 1893 on the basis of his contributions in applied mathematics (statistical mechanics and hydrodynamics), geometry, and the theory of functions. Although he continued to contribute to these areas throughout his working life, and added probability theory to his fields of interest during World War I, he turned to the theory of groups in 1893, and it is for his discoveries in this subject that he is remembered.
Burnside treated every aspect of the theory of finite groups. He was much concerned with the search for finite simple groups, and made the famous conjecture, finally proved by Walter Feit and John Thompson in 1962, that there are no simple groups of odd composite order (see THE CLASSIFICATION OF FINITE SIMPLE GROUPS [V.7]). He helped to develop character theory, which had been created by FROBENIUS [VI.58] in 1896, into a tool for proving theorems of pure group theory, using it in 1904 to spectacular effect when he proved his so-called Pα qβ-theorem: the theorem that groups whose orders are divisible by at most two different prime numbers are soluble. By asking, in effect, whether a group all of whose elements have finite order and which is generated by finitely many elements must be finite, he launched the huge area of research which for much of the twentieth century was known as the Burnside problem (see GEOMETRIC AND COMBINATORIAL GROUP THEORY [IV.10 §5.1]).
Although CAYLEY [VI.46] and the Reverend T. P. Kirkman had written about groups before him, he was the only British mathematician to work in group theory until Philip Hall started his mathematical career in 1928. Burnside’s influential book Theory of Groups of Finite Order (1897) was written in the hope of “arousing interest among English mathematicians in a branch of pure mathematics which becomes the more interesting the more it is studied.” Its influence in his own country was minimal, however, until several years after his death. It went to a second edition in 1911 (reprinted 1955), which differs from the first in that it has been substantially revised and, in particular, it includes chapters about the character theory of finite groups and its applications—mathematics which had been much developed by Frobenius, Burnside, and Schur over the fifteen years following the invention of character theory in 1896.
Further Reading
Curtis, C. W. 1999. Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. Providence, RI: American Mathematical Society.
Neumann, P. M., A. J. S. Mann, and J. C. Tompson. 2004. The Collected Papers of William Burnside, two volumes. Oxford: Oxford University Press.