威廉·伯恩赛德William Burnside

威廉·伯恩赛德的父母是Emma Knight和伯恩赛德。老伯恩赛德有苏格兰血统，他的祖父从苏格兰搬到伦敦，在那里他是书商Seeley和伯恩赛德的合伙人。老伯恩赛德是一位商人，住在Paddington的Howley Place 7号，他的父母两个儿子中的长子伯恩赛德就出生在那里。然而，到六岁时Willian成了孤儿[1]：-

伯恩赛德在Christ's Hospital接受教育——当时位于Newgate Street——并在文法学校和数学学校都取得了优异成绩。

基督公学是一所招收父母无力支付寄宿学校学费的男孩的学校，因此对像伯恩赛德这样的孤儿来说特别合适。他于1871年10月获得奖学金进入剑桥大学圣弗瑞兹·约翰学院。1873年，他从圣约翰学院转到彭布罗克学院，并非出于学术原因，而是因为圣约翰学院拥有如此出色的划船队，以至于伯恩赛德不够优秀，无法进入他们的第一船队。他能够进入彭布罗克的第一船队，所以他转到了那里，于1875年毕业，获得第二名数学荣誉学位考试一等及格者（Wrangler），与乔治·克里斯托并列。然而，伯恩赛德被认为拥有最优美的数学风格。他在剑桥的老师包括应用数学方面的乔治·加布里埃尔·斯托克斯、约翰·柯西·亚当斯和詹姆斯·克拉克·麦克斯韦，以及纯数学方面的阿瑟·凯莱。他们对伯恩赛德的研究方向产生了重大影响。他首先获得史密斯奖，并于1875年获得彭布罗克学院的研究员职位，从1875年到1886年担任该职位，并成为学院讲师。事实上，彭布罗克有着应用数学的传统，因此因划船而做出的决定在很大程度上决定了伯恩赛德在数学教学和研究中的方向。

他讲授流体动力学，但他1883年发表的第一篇论文研究的是椭圆函数。1885年，他被任命为格林尼治皇家海军学院的数学教授，此后他的研究也转向了流体动力学。伯恩赛德在流体动力学方面的大部分工作都涉及复变量的使用，在1891年和1892年的论文中，他考虑了复变量线性分式变换的群。他的工作很快转向群的研究，从1894年起，他几乎完全专注于群论的研究。

我们提到过，伯恩赛德是一位出色的桨手，是彭布罗克学院的“7”号并担任队长，但他被认为体重太轻，无法进入大学赛艇队，因此从未获得赛艇蓝。然而，他始终对赛艇保持兴趣，后来也对钓鱼产生兴趣。他一直热爱苏格兰，一生中不断去那里钓鱼度假。1886年12月25日，他刚接任格林尼治的讲席不久，便与Alexandrina Urquhart结婚。Alexandrina是苏格兰罗斯郡Poolewe一位佃农的女儿。他们育有两个儿子和三个女儿。回顾伯恩赛德的职业生涯，也许最令人惊讶的是他拒绝了彭布罗克学院请他回母校的邀请，宁愿留在格林尼治。事实上，他两次拒绝了彭布罗克学院的邀请，因为在乔治·加布里埃尔·斯托克斯于1903年去世后，学院又邀请伯恩赛德出任学院院长。他再次选择拒绝这一显赫的职位。

伯恩赛德于1893年因在流体动力学和复函数理论方面的工作而当选为皇家学会会士。然而，正是在1893年，他发表了他的第一篇关于有限单群的论文，表明交错群$A_{5}$是唯一阶为四个（不一定不同）素数乘积的有限单群。这篇论文是一系列论文中的第一篇，伯恩赛德自己描述如下（例如见[3]）：-

它们主要关注某些检验的证明，这些检验可以在特定情况下应用，以确定给定阶的单群是否可能存在。

例如，他在1895年发表的一篇论文中证明，如果一个偶数阶群有一个循环彼得·卢德维格·梅德尔·西罗2-子群，那么该群不可能是单群。他在群论方面的工作迅速进展，并于1897年出版了The Theory of Groups of Finite Order，这是第一部英文群论专著。他在引言中写道：-

本专著旨在向读者介绍有限阶群论的主要轮廓，而不涉及任何应用。这个主题在我国迄今很少受到关注；如果通过这本书，我能成功地在英国数学家中引起对纯数学一个分支的兴趣，这个分支越研究越迷人，我将感到非常满意。

这本书对群论的发展产生了重大影响。1899年，伯恩赛德当选为伦敦数学会理事会成员，同年该学会授予他奥古斯塔斯·德摩根奖章。他从1906年到1908年担任该学会主席，并继续在理事会任职直到1917年。他在1908年任期结束时发表了关于有限群的主席演讲，但仍然觉得英国数学家对此兴趣不大：-

有人向我建议，我应该利用当前的机会，介绍一下有限群论的最新进展。……但是……在当前场合，我若试图介绍该理论的最新进展……对相当一部分听众来说肯定毫无趣味。毫无疑问，有限群论至今未能引起除极少数英国数学家之外的任何人的兴趣；与欧洲大陆和美国对该主题投入的关注相比，英国这种兴趣的缺乏在我看来非常引人注目。我打算将我的演讲用于思考此地与他处对该主题投入关注量的显著差异，并尝试解释这一差异。

现在让我们考察伯恩赛德对群论的更多贡献。费迪南德·格奥尔格·弗罗贝尼乌斯于1896年开始发展群的表示论和特征标理论。伯恩赛德很快认识到费迪南德·格奥尔格·弗罗贝尼乌斯方法的重要性，并开始使用特征标理论。他最重要的结果之一，即阶为$p^{m}q^{n}$的群是可解的，出现于1904年。这一结果的特例已由彼得·卢德维格·梅德尔·西罗（1872年的$n = 0$情形）、费迪南德·格奥尔格·弗罗贝尼乌斯（1895年的$n = 1$情形）和卡米耶·若尔当（1898年的$n = 2$情形）证明。伯恩赛德猜想每个奇数阶有限群都是soluble，他未能证明这一结果并不奇怪，因为直到1962年瓦尔特·法伊特和约翰·格里格斯·汤普森才在一篇300页的论文中证明了该结果。

今天群论的许多方面仍然沿着伯恩赛德所设定的方向发展。他关于当元素具有固定有限阶时群的有限性的著名“伯恩赛德问题”，至今仍是群论研究的一个主要领域。事实上，1994年Fields Medallist叶菲姆·泽尔曼诺夫因与伯恩赛德问题相关的工作而被授予奖章。

如果说The Theory of Groups of Finite Order的第一版很重要，那么1911年出版的第二版则是一部经典，至今仍被广泛阅读，其中系统地发展了该主题，包括费迪南德·格奥尔格·弗罗贝尼乌斯的特征标理论和伯恩赛德使用这些方法的工作。他在序言中写道：-

自本书第一版问世以来，有限群论取得了非常可观的进展。特别是线性替换群论已成为若干作者众多重要研究的主题；原序中省略对其任何叙述的理由已不再成立。事实上，现在可以说，要取得抽象理论的进一步进展，必须主要着眼于将群表示为线性替换群。

伯恩赛德一生中发表了约150篇论文，其中约50篇关于群论。事实上，在他生命的最后几年，他转向了概率论，其关于该主题的第一篇论文出现于1918年。他留下了一部关于概率的完整书稿，在他去世后的第二年以The Theory of Probability出版。

1925年12月22日，伯恩赛德轻微中风，正如他在次年1月19日给Baker的信中所解释的那样：-

我在12月22日轻微中风，虽然恢复得很好，但还远未脱离医生的照管。除其他事项外，他禁止我对数学感兴趣。

伯恩赛德确实回到了他的数学研究，并在当年晚些时候发表了On a group of order 25920 and the projective transformations of a cubic surface，但他于1927年去世。去世前，他回复了菲利浦·霍尔，后者写信向他征求最有益于研究的群论问题的建议。菲利浦·霍尔后来证明是伯恩赛德非常合格的继任者，成为英格兰群论的推动者。

William Burnside's parents were Emma Knight and William Burnside. William Burnside Senior was of Scottish ancestry, his grandfather having moved from Scotland to London where he was a partner in the booksellers Seeley and Burnside. William Burnside Senior was a merchant who lived at 7 Howley Place, Paddington, where William, the elder of his parents two sons, was born. However, by the age of six Willian was an orphan [1]:-

Burnside was educated at Christ's Hospital - then situated in Newgate Street - and achieved distinction in both the grammar school and the mathematical school.

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. He entered St John's College, Cambridge in October 1871 having won a scholarship. 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. He could make the first boat for Pembroke so he moved there, graduating in 1875 as second wrangler, bracketed with George Chrystal. Burnside was, however, considered to have the most elegant mathematical style. 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. He was first Smith's Prizeman and 1875, was awarded a fellowship at Pembroke College which he held from 1875 to 1886, and became a College lecturer. In fact 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.

He lectured on hydrodynamics but his first paper, published in 1883, considered elliptic functions. After 1885, the year he was appointed professor of mathematics at the Royal Naval College at Greenwich, his research too turned towards 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.

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. However, he retained an interest in rowing and later in fishing. He always had a love for Scotland and continued to take fishing holidays there throughout his life. He married Alexandrina Urquhart on 25 December 1886 soon after he took up the Chair at Greenwich. Alexandrina was the daughter of a crofter Poolewe in the county of Ross, Scotland. They had 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. In fact 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. Again he chose to turn down this prestigious offer.

Burnside was elected a Fellow of the Royal Society in 1893 for his work on hydrodynamics and complex function theory. However it was in 1893 that he published his first paper on finite simple groups, showing that the alternating group $A_{5}$ is the only finite simple group whose order is the product of four (not necessarily distinct) primes. This paper was the first of a series which Burnside described himself as follows (see for example [3]):-

They are concerned chiefly with the proof of certain tests that may be applied in particular cases to determine whether it is possible for a simple group of a given order to exist.

For example he proved in a paper published in 1895 that if a group of even order has a cyclic Sylow 2-subgroup then the group cannot be simple. 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. He wrote in the Introduction:-

The present treatise is intended to introduce to the reader the main outlines of the theory of groups of finite order apart from any applications. The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.

This book 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 1906 to 1908 and continued to serve on the Council until 1917. He gave his presidential address at the end of his term of office in 1908 on finite groups but still felt that there was little interest among British mathematicians:-

It has been suggested to me that I should take advantage of the present occasion to give an account of the recent progress of the theory of groups of finite order. ... But ... any attempt on my part to give, on the present occasion, an account of the recent advance in the theory ... would certainly be uninteresting to a considerable number of my audience. It is undoubtedly the fact that the theory of groups of finite order has failed, so far, to arouse the interest of any but a very small number of English mathematicians; and this want of interest in England, as compared with the amount of attention devoted to the subject both on the Continent and in America, appears to me very remarkable. I propose to devote my address to a consideration of the marked difference in the amount of attention devoted to the subject here and elsewhere, and to some attempt to account for this difference.

Let us now examine some more of Burnside's contributions to group theory. 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 $p^{m}q^{n}$ 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 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. He wrote in the Preface:-

Very considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. in fact it is now true to say that for further advances in the abstract theory one must look largely to the representation of a group as a group of linear substitutions.

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.

On 22 December 1925 Burnside had a slight stroke as he explained in a letter to Baker on 19 January of the following year:-

I had a slight stroke on 22 December and though I have gone on very well I am by no means out of the doctor's hands yet. Among other things he forbids is an interest in mathematics.

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. Before he died he replied to Philip Hall who wrote to him asking for suggestions for the most profitable group theory problems to study. Hall was to prove a very worthy successor to Burnside as the promoter of group theory in England.