##
**The collected papers of William Burnside. Vol. 1: Commentary on Burnside’s life and work; papers 1883–1899. Vol. 2: 1900-1926. Edited by Peter M. Neumann, A. J. S. Mann and Julia C. Tompson.**
*(English)*
Zbl 1060.20001

Oxford: Oxford University Press (ISBN 0-19-850585-X/hbk set; 0-19-850586-8/v.1; 0-19-85087-6/v.2). 1584 p. (2004).

William Burnside (1852-1927) is recognized as one of the three founding fathers of group representation theory, the other two being Georg Frobenius and Issai Schur. The collected papers of Frobenius were published in 1968 and those of Schur in 1975. Taking into account the three volumes of collected papers of Richard Brauer, which appeared in 1980, the publication of The Collected Papers of William Burnside provides us with easy access to the work of all four of these great pioneers and we can now study their most famous contributions to mathematics in the form in which they were originally written. Furthermore, we can also read their less well known work and see if they anticipated future developments in the subject.

The Collected Papers under review here reveal that Burnside was a versatile mathematician. He is best known for his group-theoretic researches, but he published papers in applied mathematics, algebraic geometry, theory of automorphic functions and number theory. It should be pointed out that Burnside was essentially self-taught as an algebraist, since his mathematical training at Cambridge University emphasized the traditional curriculum of mathematical physics, which had become dominant in the 19th century, and his career from 1885 was spent at the Royal Naval College, Greenwich as a teacher of applied subjects such as hydrodynamics. Charles W. Curtis’s book Pioneers of Representation Theory: Frobenius, Burnside, Schur and Brauer (1999) provides the interested reader with biographical detail about Burnside and gives a good analysis of his work, especially as it relates to the parallel development of group character theory at his hands and those of Frobenius. For this reason, we will confine our review to a discussion of some aspects of Burnside’s research which are perhaps less well known, but are made available by a study of the Collected Papers.

Burnside discovered in himself a remarkable affinity for group theory, beginning with his first paper on the subject, published in 1893. The evidence of the Collected Papers suggests that his interest in Riemann surfaces, elliptic functions and the modular group led him to embark on the study of a topic hitherto little appreciated in his home country (we exclude Cayley from this assessment, but his group-theoretic papers had little impact among Cambridge mathematicians). Only four years after his introduction to group theory, he had sufficiently mastered the subject to write the influential treatise The Theory of Groups of Finite Order (1897), a work considerably augmented in its second edition (1911).

Burnside did not construct elaborate new theories, rather, he was driven to solve particular problems, using or modifying existing techniques. We will describe here a few of his papers which display insight into subjects subsequently studied intensively by later generations of mathematicians.

\(\bullet\) On a class of groups of finite order (1899). In this paper, Burnside considered the finite groups \(G\) of even order in which each element either has odd order or order 2. He showed that \(G\) is either a Frobenius group whose kernel is an elementary Abelian 2-group, a Frobenius group whose kernel is an Abelian group of odd order and whose complement has order 2, or \(G\) is isomorphic to the simple group \(\text{SL}(2,2^n)\), where \(n\geq 2\). This is the first example of a type of classification problem which became popular in the 1950’s. Burnside’s work seems to have passed unnoticed and his main result was rediscovered in the 1952 thesis of K. A. Fowler, written at the University of Michigan under the direction of Richard Brauer. The key idea for Burnside was that two involutions generate a dihedral group, a technique that became a cornerstone of the “involution methods” developed by Brauer. Burnside had already used such involution methods to prove that a Frobenius group whose complement has even order has an Abelian kernel.

\(\bullet\) On the arithmetical nature of the coefficients in a group of linear substitutions of finite order (1905). From a modern point of view, the paper concerns Schur index problems. Specifically, Burnside asked when a representation of a group can be realized over the same field as its character field. He certainly did not answer this question, which is of course too complicated, but was led to consider finite linear groups in which no non-identity element fixes a non-zero vector. Such groups are precisely the class of Frobenius complements, a topic of much interest to Burnside. He made attempts to classify such groups but did not obtain a definitive list of examples. Nonetheless, the paper shows Burnside’s early appreciation of a delicate question.

\(\bullet\) The determination of all groups of rational linear substitutions of finite order which contain the symmetric group in the variables (1911). The starting point is the group \(G_n\) of \(n\times n\) permutation groups, naturally isomorphic to the symmetric group \(S_n\). The problem is to find all finite groups \(G\) of \(n\times n\) matrices with rational coefficients which contain the subgroup \(G_n\). Except when \(n=6\), 7 or 8, Burnside showed that either \(G\) contains \(G_n\) as a direct factor or else \(G\) is a known type of group related to \(S_{n+1}\) or the group of signed permutation matrices, of order \(2^nn!\). In the exceptional cases, \(G\) may be the Weyl group of type \(E_6\), \(E_7\) or \(E_8\), occurring in dimensions \(n=6\), \(7\) and \(8\). The paper contains some remarkable calculations in which Burnside relates the combinatorics of the exceptional groups to geometrical configurations involving the intersection of systems of spheres. Burnside anticipated much of the 20th century interest in Weyl groups (groups generated by real orthogonal reflections) in this work. Coxeter remarks in his book Regular Polytopes that Burnside had read part of a paper by Thorold Gosset on polytopes in 1897 but probably did not see later any connection between Gosset’s work and his own 1911 paper.

\(\bullet\) The condition that an irreducible group of linear substitutions on \(n\) variables may contain a substitution with \(n-1\) unit multipliers (1911). Probably motivated by the paper described above, Burnside investigated irreducible finite complex linear groups containing a unitary reflection \(\tau\) of order \(m\geq 2\). He showed that unless \(m=2\), 3 or 5, \(\tau\) commutes with its own conjugates and so lies in a normal Abelian subgroup. The corresponding linear group is imprimitive in the non-exceptional cases. The subject of complex linear groups generated by unitary reflections has become important in group theory and algebraic geometry since the appearance of the important paper Finite unitary reflection groups by G. C. Shephard and J. A. Todd (1954).

We hope that this selection of topics gives some indication of Burnside’s good taste and instinctive appreciation of important mathematical structures.

With the exception of his textbooks, the Collected Papers has brought together virtually everything Burnside wrote, including book reviews, presidential addresses and encyclopaedia articles. In addition to Burnside’s own output, the editors have assembled a commentary on Burnside’s life and work, occupying more than 100 pages in the first volume. There is a foreword by Walter Feit, who unfortunately died just before the work was published, and a bibliography of Burnside’s writing, taken from an article by A. Wagner and V. Rosenthal. Peter Neumann, one of the editors, has written an authoritative essay entitled The context of Burnside’s contributions to group theory, which discusses Burnside’s work and its relation to earlier and later research in group theory. Michael Newman has provided an essay A still unsettled question concerning the Burnside problem for periodic groups. In Burnside and finite simple groups, Ronald Solomon has evaluated Burnside’s work in the light of the methods employed in the classification of finite simple groups. Charles Curtis has contributed an essay on Burnside’s research on representation theory, based partly on his book mentioned above. In Burnside’s applied mathematics, June Barrow-Green gives an appraisal of Burnside’s work in mathematical physics, which was an important aspect of how he earned his living but is largely forgotten today. The final essay Notes on Burnside’s life, by Martin Everett, Anthony Mann and Kathy Young, has uncovered many new aspects of Burnside’s life and the lives of his family members, including several photographs.

In conclusion, we must congratulate the editors, Peter Neumann, Anthony Mann and Julia Tompson, for bringing this valuable project to fruition, and thereby doing justice to the memory of a man who discovered many of the basic working methods of group theory. Burnside’s achievements were not greatly appreciated in his lifetime but, with the help of these Collected Papers and Curtis’s book, this state of affairs is changing, and mathematicians with a taste for the historical development of a major discipline now have the opportunity to see how Burnside’s ideas evolved and how he anticipated some of the key concepts of 20th century group theory.

The Collected Papers under review here reveal that Burnside was a versatile mathematician. He is best known for his group-theoretic researches, but he published papers in applied mathematics, algebraic geometry, theory of automorphic functions and number theory. It should be pointed out that Burnside was essentially self-taught as an algebraist, since his mathematical training at Cambridge University emphasized the traditional curriculum of mathematical physics, which had become dominant in the 19th century, and his career from 1885 was spent at the Royal Naval College, Greenwich as a teacher of applied subjects such as hydrodynamics. Charles W. Curtis’s book Pioneers of Representation Theory: Frobenius, Burnside, Schur and Brauer (1999) provides the interested reader with biographical detail about Burnside and gives a good analysis of his work, especially as it relates to the parallel development of group character theory at his hands and those of Frobenius. For this reason, we will confine our review to a discussion of some aspects of Burnside’s research which are perhaps less well known, but are made available by a study of the Collected Papers.

Burnside discovered in himself a remarkable affinity for group theory, beginning with his first paper on the subject, published in 1893. The evidence of the Collected Papers suggests that his interest in Riemann surfaces, elliptic functions and the modular group led him to embark on the study of a topic hitherto little appreciated in his home country (we exclude Cayley from this assessment, but his group-theoretic papers had little impact among Cambridge mathematicians). Only four years after his introduction to group theory, he had sufficiently mastered the subject to write the influential treatise The Theory of Groups of Finite Order (1897), a work considerably augmented in its second edition (1911).

Burnside did not construct elaborate new theories, rather, he was driven to solve particular problems, using or modifying existing techniques. We will describe here a few of his papers which display insight into subjects subsequently studied intensively by later generations of mathematicians.

\(\bullet\) On a class of groups of finite order (1899). In this paper, Burnside considered the finite groups \(G\) of even order in which each element either has odd order or order 2. He showed that \(G\) is either a Frobenius group whose kernel is an elementary Abelian 2-group, a Frobenius group whose kernel is an Abelian group of odd order and whose complement has order 2, or \(G\) is isomorphic to the simple group \(\text{SL}(2,2^n)\), where \(n\geq 2\). This is the first example of a type of classification problem which became popular in the 1950’s. Burnside’s work seems to have passed unnoticed and his main result was rediscovered in the 1952 thesis of K. A. Fowler, written at the University of Michigan under the direction of Richard Brauer. The key idea for Burnside was that two involutions generate a dihedral group, a technique that became a cornerstone of the “involution methods” developed by Brauer. Burnside had already used such involution methods to prove that a Frobenius group whose complement has even order has an Abelian kernel.

\(\bullet\) On the arithmetical nature of the coefficients in a group of linear substitutions of finite order (1905). From a modern point of view, the paper concerns Schur index problems. Specifically, Burnside asked when a representation of a group can be realized over the same field as its character field. He certainly did not answer this question, which is of course too complicated, but was led to consider finite linear groups in which no non-identity element fixes a non-zero vector. Such groups are precisely the class of Frobenius complements, a topic of much interest to Burnside. He made attempts to classify such groups but did not obtain a definitive list of examples. Nonetheless, the paper shows Burnside’s early appreciation of a delicate question.

\(\bullet\) The determination of all groups of rational linear substitutions of finite order which contain the symmetric group in the variables (1911). The starting point is the group \(G_n\) of \(n\times n\) permutation groups, naturally isomorphic to the symmetric group \(S_n\). The problem is to find all finite groups \(G\) of \(n\times n\) matrices with rational coefficients which contain the subgroup \(G_n\). Except when \(n=6\), 7 or 8, Burnside showed that either \(G\) contains \(G_n\) as a direct factor or else \(G\) is a known type of group related to \(S_{n+1}\) or the group of signed permutation matrices, of order \(2^nn!\). In the exceptional cases, \(G\) may be the Weyl group of type \(E_6\), \(E_7\) or \(E_8\), occurring in dimensions \(n=6\), \(7\) and \(8\). The paper contains some remarkable calculations in which Burnside relates the combinatorics of the exceptional groups to geometrical configurations involving the intersection of systems of spheres. Burnside anticipated much of the 20th century interest in Weyl groups (groups generated by real orthogonal reflections) in this work. Coxeter remarks in his book Regular Polytopes that Burnside had read part of a paper by Thorold Gosset on polytopes in 1897 but probably did not see later any connection between Gosset’s work and his own 1911 paper.

\(\bullet\) The condition that an irreducible group of linear substitutions on \(n\) variables may contain a substitution with \(n-1\) unit multipliers (1911). Probably motivated by the paper described above, Burnside investigated irreducible finite complex linear groups containing a unitary reflection \(\tau\) of order \(m\geq 2\). He showed that unless \(m=2\), 3 or 5, \(\tau\) commutes with its own conjugates and so lies in a normal Abelian subgroup. The corresponding linear group is imprimitive in the non-exceptional cases. The subject of complex linear groups generated by unitary reflections has become important in group theory and algebraic geometry since the appearance of the important paper Finite unitary reflection groups by G. C. Shephard and J. A. Todd (1954).

We hope that this selection of topics gives some indication of Burnside’s good taste and instinctive appreciation of important mathematical structures.

With the exception of his textbooks, the Collected Papers has brought together virtually everything Burnside wrote, including book reviews, presidential addresses and encyclopaedia articles. In addition to Burnside’s own output, the editors have assembled a commentary on Burnside’s life and work, occupying more than 100 pages in the first volume. There is a foreword by Walter Feit, who unfortunately died just before the work was published, and a bibliography of Burnside’s writing, taken from an article by A. Wagner and V. Rosenthal. Peter Neumann, one of the editors, has written an authoritative essay entitled The context of Burnside’s contributions to group theory, which discusses Burnside’s work and its relation to earlier and later research in group theory. Michael Newman has provided an essay A still unsettled question concerning the Burnside problem for periodic groups. In Burnside and finite simple groups, Ronald Solomon has evaluated Burnside’s work in the light of the methods employed in the classification of finite simple groups. Charles Curtis has contributed an essay on Burnside’s research on representation theory, based partly on his book mentioned above. In Burnside’s applied mathematics, June Barrow-Green gives an appraisal of Burnside’s work in mathematical physics, which was an important aspect of how he earned his living but is largely forgotten today. The final essay Notes on Burnside’s life, by Martin Everett, Anthony Mann and Kathy Young, has uncovered many new aspects of Burnside’s life and the lives of his family members, including several photographs.

In conclusion, we must congratulate the editors, Peter Neumann, Anthony Mann and Julia Tompson, for bringing this valuable project to fruition, and thereby doing justice to the memory of a man who discovered many of the basic working methods of group theory. Burnside’s achievements were not greatly appreciated in his lifetime but, with the help of these Collected Papers and Curtis’s book, this state of affairs is changing, and mathematicians with a taste for the historical development of a major discipline now have the opportunity to see how Burnside’s ideas evolved and how he anticipated some of the key concepts of 20th century group theory.

Reviewer: Roderick Gow (Dublin)

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

20-03 | History of group theory |

20C15 | Ordinary representations and characters |