Representations of Finite Groups: a Hundred Years, Part II T

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Representations of Finite Groups: A Hundred Years, Part II T. Y. Lam

Recapitulation trices. This work led Frobenius to formulate, in The origin of the representation theory of finite 1897, the modern definition of a (matrix) repre- groups can be traced back to a correspondence be- sentation of a group G as a homomorphism tween R. Dedekind and F. G. Frobenius that took D : G GLn(C) (for some n). With this definition → place in April of 1896. The present article is based in place, the character χD : G C of the repre- → on several lectures given by the author in 1996 in sentation is simply defined by χD(g) = trace(D(g)) commemoration of the centennial of this occa- (for every g G). The idea of studying a group ∈ sion. through its various representations opened the In Part I of this article we recounted the story door to a whole new direction of research in group of how Dedekind proposed to Frobenius the prob- theory and its applications. lem of factoring a certain homogeneous polyno- Having surveyed Frobenius’s invention of char- mial arising from a determinant (called the “group acter theory and his subsequent monumental con- determinant”) associated with a finite group G. In tributions to representation theory in Part I of this the case when G is abelian, Dedekind was able to article, we now move on to tell the story of another factor the group determinant into linear factors giant of the subject, the English group theorist using the characters of G (namely, homomor- W. Burnside. This comprises Part II of the article, phisms of into the group of nonzero complex G which can be read largely independently of Part I. numbers). In a stroke of genius, Frobenius invented For the reader’s convenience, the few biblio- a general character theory for arbitrary finite graphical references needed from Part I are re- groups, and used it to give a complete solution to produced here, with the same letter codes for the Dedekind’s group determinant problem. Interest- sake of consistency. As in Part I, [F: (53)] refers to ingly, Frobenius’s first definition of (nonabelian) characters was given in a rather ad hoc fashion, via paper (53) in Frobenius’s collected works [F]. Burn- the eigenvalues of a certain set of commuting ma- side’s papers are referred to by the year of publi- cation, from the master list compiled by Wagner T. Y. Lam is professor of mathematics at the University and Mosenthal in [B]. Consultation of the original of California, Berkeley. His e-mail address is papers is, however, not necessary for following [email protected]. the general exposition in this article. This article, in two parts, is a slightly abridged version of an article appearing concurrently in the Proceedings of William Burnside (1852–1927) the “Aspects of Mathematics” conference (N. Mok, ed.), Remarks in this section about Burnside’s life and University of Hong Kong, H.K., 1998. Part I, concerning work are mainly taken from A. R. Forsyth’s obitu- the study of group determinants by Ferdinand Georg Frobenius and the resulting theory of characters and ary note [Fo] on Burnside published in the Journal group representations, appeared in the March 1998 issue of the London Mathematical Society the year after of the Notices, pages 361–372. Burnside’s death, and from the forthcoming book

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of C. Curtis on theory, differential equations, etc.) to topics in the pioneers of theoretical physics such as kinematics, elasticity, representation electrostatics, hydrodynamics, and the theory of theory [Cu 2, Ch. gases. So not surprisingly, in the first fifteen years 3]. of his career, Burnside’s published papers were ei- Born in Lon- ther in these applied areas, or else in elliptic and don of Scottish automorphic functions and differential geometry. stock, William On account of this work, he was elected Fellow of Burnside re- the Royal Society in 1893. Coincidentally, it was also ceived a tradi- around this time that Burnside’s mathematical in- tional university terests began to shift to group theory, a subject to education in St. which he was to devote his main creative energy John’s and Pem- in his mature years. broke Colleges in After authoring a series of papers entitled “Notes Cambridge. In on the theory of groups of finite order” (and oth- Pembroke he dis- ers), Burnside published his group theory book tinguished him- [B 1] in 1897, the first in the English language of- self both as a fering a comprehensive treatment of finite group mathematician theory. A second, expanded edition with new ma- and as an oars- terial on group representations appeared in 1911. man, graduating For more than half a century, this book was with- from Cambridge out doubt the one most often referred to for a de- as Second Wran- tailed exposition of basic material in group theory. gler in the 1875 Reprinted by Dover in 1955 (and sold for $2.45),

Photograph courtesy of the Institut Mittag-Leffler. Mathematical Burnside’s book is now enshrined as one of the true Tripos. He took William Burnside. classics of mathematics. We will have more to say up a lectureship about this book in the next section.2 in Cambridge after that, and remained there for some ten years, Since group theory was not a popular subject teaching mathematics and acting as coach for both in England at the turn of the century, Burnside’s the Math Tripos and for the rowing crews. In 1885, group-theoretic work was perhaps not as much ap- at the instance of the Director of Naval Instruction preciated as it could have been. When Burnside died (a former Cambridge man named William Niven), in 1927, the London Times reported the passing of Burnside accepted the position of professor of “one of the best known Cambridge athletes of his 3 mathematics in the Royal Naval College at Green- day”. We can blame this perhaps on the journal- wich. He spent the rest of his career in Greenwich, ist’s ignorance and lack of appreciation of math- but kept close ties with the Cambridge circles, and ematics. However, even in Forsyth’s detailed obit- never ceased to take an active role in the affairs uary, which occupied seventeen pages of the of the London Mathematical Society, serving long Journal of the London Math Society, no more than terms on its Council, including a two-year term as a page was devoted to Burnside’s work in group president (1906–08). In Greenwich he taught math- theory, even though Forsyth was fully aware that ematics to naval personnel, which included gun- it was this work that would “provide the most con- nery and torpedo officers, civil and mechanical tinuous and most conspicuous of his contribu- engineers, as well as cadets. The teaching task was tions to his science.” None of Burnside’s greatest not too demanding for Burnside, which was just achievements that now make him a household fine, as it afforded him the time to pursue an active program of research. Although physically away 2Those familiar with Dover publications will know that from the major mathematical centers of England, there are two other Dover reprints of books by “Burnside”: he kept abreast of the current progress in research one is Theory of Probability, and the other is Theory of throughout his career, and published a total of Equations. The former was indeed written by William some 150 papers in pure and applied mathemat- Burnside; published posthumously in 1928, it was also one of the earliest texts in probability theory written in Eng- ics. By all accounts, Burnside led a life of steadfast lish. However, the 2-volume work Theory of Equations was devotion to his science. written (ca. 1904) by Panton and another Burnside. William Burnside’s early training was very much steeped Snow Burnside, professor of mathematics in Dublin, was in the tradition of applied mathematics in Cam- a contemporary of William Burnside; they published pa- bridge.1 At that time, applied mathematics meant pers in the same English journals, one as “W. S. Burnside” essentially the applications of analysis (function and the other simply as “W. Burnside”. An earlier com- mentary on this was given by S. Abhyankar [Ab, footnote 1According to Forsyth [Fo], pure mathematics was then 43, p. 91]. largely “left to Cayley’s domain, unfrequented by aspirants 3The full text of the London Times obituary on Burnside for high place in the tripos.” was quoted in [Cu 2, Ch. 3].

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name in group theory was even mentioned in that Burnside realized that these results were not Forsyth’s article. I can think of two reasons for this. in their final form, for he wrote in [B 1, 1st ed., The first is perhaps that Forsyth did not have any p. 344]: real appreciation of group theory. While he was If the results appear fragmentary, it Sadlerian Professor of Mathematics in Cambridge, must be mentioned that this branch of his major field was function theory and differen- the subject has only recently received tial equations.4 We cannot blame him for being attention: it should be regarded as a more enthusiastic about Burnside’s work in func- promising field of investigation than tion theory and applied mathematics; after all, it as one which is thoroughly explored. was this work that won Burnside membership in the Royal Society. Second, Burnside’s achievements Burnside was right on target in his perception that in group theory were truly way ahead of his time; much more was in store for this line of research. the deep significance of his ideas and the true However, Burnside’s assessment at that time of the power of his vision only became clear many a year possible role of groups of linear substitutions was after his death. Today, I do not hear my applied a bit tentative. Since the theory of permutation math colleagues talk about Burnside’s work in hy- groups occupied a large part of [B 1] while groups drodynamics or the kinetic theory of gases, but I of linear substitutions hardly received any atten- will definitely teach my students Burnside’s great tion, Burnside felt obliged to give an explanation proof of the paqb theorem (that any group of to his readers. In the preface to the first edition of order paqb is solvable) in my graduate course in [B 1], he wrote: group representation theory! In mathematics, as My answer to this question is that while, in other sciences, it is time that will tell what are in the present state of our knowledge, the best and the most lasting human accomplish- many results in the pure theory are ar- ments. rived at most readily by dealing with properties of substitution groups, it Theory of Groups of Finite Order (1897, would be difficult to find a result that 1911) could be most directly obtained by the Through his work on the automorphic functions consideration of groups of linear trans- of Klein and Poincaré, Burnside was knowledgeable formations. about the theory of discontinuous groups. It was perhaps this connection that eventually steered him Little did he know that, just as his book was going away from applied mathematics and toward re- to press, Frobenius on the continent had just made search on the theory of groups of finite order. In his breakthroughs in the invention of group char- the early 1890s, Burnside followed closely Hölder’s acters, and was in fact writing up his first mem- work on groups of specific orders; soon he was pub- oir [F: (56)] on the new representation theory of lishing his own results on the nature of the order groups! As it turned out, the next decade wit- of finite simple groups. Frobenius’s early papers nessed some of the most spectacular successes in applying representation theory to the study of the in group theory apparently first aroused his interest structure of finite groups—and Burnside himself in finite solvable groups.5 was to be a primary figure responsible for these The first edition of Burnside’s masterpiece The- successes. There was no question that Burnside ory of Groups of Finite Order appeared in 1897; it wanted his readers to be brought up to date on this was clearly the most important book in group the- exciting development. When the second edition of ory written around the turn of the century. While [B ] came out in 1911 (fourteen years after the intended as an introduction to finite group theory 1 first), it was a very different book, with much more for English readers, the book happened to contain definitive results and with six brand new chapters some of the latest research results in the area at introducing his readers to the methods of group that time. For instance, groups of order paqb were representation theory! In the preface to this new shown to be solvable if either a 2 or if the Sylow ≤ edition, Burnside wrote: groups were abelian, and (nonabelian) simple groups were shown to have even order if the order In particular the theory of groups of lin- was the product of fewer than six primes.6 It is clear ear substitutions has been the subject of numerous and important investiga- 4His 2-volume work on the former and 6-volume work on tions by several writers; and the reason the latter were quite popular in his day. given in the original preface for omit- 5A group G is solvable if it can be constructed from ting any account of it no longer holds abelian groups via a finite number of group extensions. good. In fact it is now more true to say In case G is finite, an equivalent definition is that the com- that for further advances in the ab- position factors of G are all of prime order. stract theory one must look largely to 6Burnside showed that the order must be 60, 168, 660, the representation of a group as a group or 1,092. of linear substitutions.

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With these words, Burnside brought the subject of ference being, that Molien worked with C G as an group theory into the twentieth century, and he associative algebra (or “hypercomplex system”) went on to present 500 pages of great mathemat- while Burnside worked with C G as a Lie algebra. ics in his elegant and masterful style. Though later However, Molien’s paper [M 1] was understood by authors have found an occasional mistake in [B1], few, which was perhaps why it did not receive the and there are certainly some typographical er- recognition that it deserved. Burnside, for one, rors,7 Burnside’s book has remained as valuable a was apparently frustrated by the exposition in reference today as it has been throughout this Molien’s paper. In a later work [B: 1902f], in re- century. I myself have developed such a fondness ferring to [M 1], Burnside lamented openly: for Burnside’s book that every time I walk into a used book store and have the good fortune to find It is not, in fact, very easy to find exactly a copy of the Dover edition on the shelf, I would what is and what is not contained in buy it. By now I have acquired seven (or is it eight?) Herr Molien’s memoir. copies. True book connoisseurs would go instead Later, the methods of both Molien and Burnside for the first edition of Burnside’s book, because of were superseded by those of Emmy Noether [N]. its scarcity and historical value. Apparently, a good As was noted in the section “Factorization of (G) copy could command a few hundred dollars in the for Modern Readers” in Part I of this article, a rare book market. quick application of the theorems of MaschkeΘ and Wedderburn à la Noether yields all there is to know Burnside’s Work in Representation Theory about representations at the basic level. After reading Frobenius’s papers [F: (53), (54)], The next stage of Burnside’s work consists of Burnside saw almost immediately the relevance of his detailed investigation into the nature of irre- Frobenius’s new theory to his own research on fi- ducible representations and their applications. Re- nite groups. What he tried to do first was to un- call that, for Frobenius, the irreducibility (or “prim- derstand Frobenius’s results in his own way. In the itivity”) of a representation was originally defined 1890s Burnside had also followed closely the work by the irreducibility of its associated determinant. of Sophus Lie on continuous groups of transfor- Since this was clearly a rather unwieldy definition, mations, so, unlike Frobenius, he was conversant a more direct alternative definition was desirable. with the methods of Lie groups and Lie algebras. Burnside [B: 1898a] and Frobenius [F: (56)] had Given a finite group G, he was soon able to define both given definitions for the irreducibility of a rep- a Lie group from G whose Lie algebra is the group resentation in terms of the representing matrices, algebra C G endowed with the bracket operation although, as Charles Curtis pointed out to me, [A, B]=AB BA. Analyzing the structure of this − these early definitions seemed to amount to what Lie algebra, he succeeded in deriving Frobenius’s we now call indecomposable representations. By principal results both on characters and on the 1898, E. H. Moore (and independently A. Loewy) had group determinant. (For more details on this, see obtained the result that any finite group of linear [Cu ] and [H ].) He published these results in sev- 2 2 substitutions admits a nondegenerate invariant eral parts, in [B: 1898a, 1900b], etc. Burnside cer- hermitian form, and in 1899, Maschke used Moore’s tainly was not claiming that he had anything new; result to prove the “splitting” of any subrepre- he wrote in [B: 1900b]: sentation of a representation of a finite group The present paper has been written (now called “Maschke’s Theorem”). With these new with the intention of introducing this results, whatever confusion that might have existed new development to English readers. It in the definition of the irreducibility of a repre- is not original, as the results arrived at sentation became immaterial. By 1901 (if not ear- are, with one or two slight exceptions, lier), Burnside was able to describe an irreducible due to Herr Frobenius. The modes of representation in no uncertain terms [B: 1900b, proof, however, are in general quite dis- p. 147]: “A group G of finite order is said to be rep- tinct from those used by Herr Frobe- resented as an irreducible group of linear substi- nius. tutions on m variables when G is simply or multiply isomorphic with the group of linear Actually, while Burnside’s methods were different substitutions, and when it is impossible to choose from Frobenius’s, what he did was close in spirit m (

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count of basic character theory in [B: 1903d], in- unity and Galois conjugates, he proved the fol- dependently of the continuous group approach lowing result in [B: 1904a]: he initiated in [B: 1898a]. One year later in [B: Theorem 4.2. Let g G and χ be an irreducible 1905b], Burnside arrived at a beautiful character- character of G. If χ(1)∈is relatively prime to the car- ization of an irreducible representation that is still dinality of the conjugacy class of g, then χ(g) is in use today: | | equal to either 0 or χ(1). Theorem 4.1. A representation D : G GLn(C) −→ Combining this result with the Second Orthog- is irreducible if and only if the matrices in D(G) onality Relation (given in the display box in Part I, span Mn(C). p. 368), he obtained a very remarkable sufficient Burnside proved this result for arbitrary (not just condition for the nonsimplicity of a (finite) group: finite) groups (and used it later in [B: 1905c] for Theorem 4.3. If a (finite) group G has a conjugacy possibly infinite groups). Subsequently, Frobenius class with cardinality pk where p is a prime and and Schur extended this theorem to “semigroups” k 1, then G is not a simple group. of linear transformations. With this hindsight, we ≥ can state Burnside’s main result in the following This sufficient condition for nonsimplicity is so powerful that, from it, Burnside obtained imme- ring-theoretic fashion: a subalgebra A of Mn(C) n diately the paqb theorem, a coveted goal of group has no nontrivial invariant subspaces in C if and theorists for more than ten years.9 It is interest- only if A = Mn(C). Stated in this form, Burnside’s result is of current interest to workers in opera- ing to point out that, in [B 1, 2nd ed., p. 323], after tor algebras and invariant subspaces. Some gen- proving the powerful theorem (4.2), Burnside simply stated the paqb theorem as “Corollary 3”: eralizations to an infinite-dimensional setting have been obtained, for instance, in Chapter 8 of [HR]. Theorem 4.4. For any primes p, q, any group G We should also point out that Burnside’s result of order paqb is solvable. holds in any characteristic; the only necessary as- The proof is so easy and pleasant by means of sumption is that the ground field be algebraically Sylow theory that we have to repeat it here. We may closed (see [L, p. 109]). assume p = q. By induction on G , it suffices to Burnside had a keen eye for the arithmetic of 6 | | show that G is not a simple group. Fix a subgroup characters (which he called “group characteris- Q of order qb (which exists by Sylow’s Theorem), tics”); many of his contributions to character the- and take an element g =1in the center of Q. If g ory were derived from his unerring sense of the is central in G, G is clearly6 nonsimple. If other- arithmetic behavior of the values of characters. The wise, CG(g) (the centralizer of g in G) is a proper following are some typical samples of his results subgroup of G containing Q. Then the conjugacy proved in this spirit: k class of g has cardinality [G : CG(g)] = p for some 1. Every irreducible character χ with χ(1) > 1 k 1, so G is again nonsimple by (4.3), as desired! has a zero value. ≥At first sight, (4.4) may not look like such a 2. The number of real-valued irreducible char- deep result. However, for many years, it defied acters of a group G is equal to the number of real the group theorists’ effort to find a purely group- conjugacy classes8 in G. theoretic proof. It was only in 1970 that, follow- 3. (Consequences of (2)) If G is even, there | | ing up on ideas of J. G. Thompson, D. Goldschmidt must exist a real-valued irreducible character other [Go] gave the first group-theoretic proof for the case than the trivial character (and conversely). If G when p, q are odd primes. Goldschmidt’s proof | | is odd, then the number of conjugacy classes in G used a rather deep result in group theory, called is congruent to G (mod 16). Glauberman’s Z(J)-theorem. A couple of years | | 4. If χ is the character of a faithful representa- later, Matsuyama [Mat] completed Goldschmidt’s tion of G, then any irreducible character is a con- work by supplying a group-theoretic proof for (4.4) stituent of some power of χ. (This result is usu- in the remaining case p =2. Slightly ahead of [Mat], ally attributed to Burnside, although, as Hawkins Bender [Be] also gave a proof of (4.4) for all p, q, pointed out in [H 3, p. 241], it was proved earlier using (among other things) a variant of the notion by Molien. A quantitative version of the result was of the “Thompson subgroup” of a p-group. While found later by R. Brauer.) these group-theoretic proofs are not long, they in- Burnside’s most lasting result in group repre- volve technical ad hoc arguments, and certainly do sentations is, of course, his great paqb theorem, not come close to the compelling simplicity and which we have already mentioned. Again, the ap- the striking beauty of Burnside’s original charac- proach he took to reach this theorem was purely ter theoretic proof. As for the more general The- arithmetic. Using arguments involving roots of 9As we indicated in the previous section, in his earlier papers Burnside had proved many special cases of this the- 8A conjugacy class is said to be real if it is closed under orem without using representation theory; much of this the inverse map. was summarized in the first edition of [B 1].

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orem 4.3, which led to the paqb theorem, a purely to the existence or nonexistence of such group-theoretic proof has not been found to date. groups would undoubtedly lead, whatever the conclusion might be, to results Burnside: Visionary and Prophet of importance; it may be recommended If one traces Burnside’s group-theoretic work back to the reader as well worth his attention. to its very beginning, it should be clear that one Also, there is no known simple group of his main objectives from the start was to un- whose order contains fewer than three derstand finite simple groups. Burnside knew the different primes.…Investigation in this role of finite simple groups from the work of Ga- direction is also likely to lead to results lois and from the Jordan-Hölder Theorem, but in of interest and importance. the 1890s, there was very little to go on. The only known finite simple groups were Galois’s alter- Thus, in one single paragraph, Burnside managed nating groups An (n 5), Jordan’s projective spe- to lay down two of the most important research ≥ cial linear groups PSL 2(p)(p 5), some of the problems in finite group theory to be reckoned with ≥ Mathieu groups, and Cole’s simple group of order in the next century. 504. The latter group is The second one was not to now recognized as remain open for very long. As PSL 2(8), but finite fields we know, Burnside’s tour de were hardly known in the “…there is no force[B: 1904a] solved this early 1890s, so in 1893, problem: the solvability of Cole [Co] had to construct known simple groups of order paqb (Theo- this group “by bare hands” group whose order rem 4.4) implied their nonsim- as a permutation group of plicity (and conversely, of degree nine.10 contains fewer course). The proof of this great By 1892, Hölder found result depended critically on all simple groups of order than three the newly invented tools of 200 . In another year, character theory. However, the ≤ with his construction of different primes.… first problem, equivalent to the the simple group of order solvability of groups of odd 504, Cole pushed Hölder’s Investigation in this order, proved to be very diffi- work to the order direction is also cult. Burnside’s odd-order pa- 660 = PSL2(11) , and in pers [B: 1900c] were clearly | | another two years, Burn- likely to lead to aimed at solving this problem, side further extended this and he obtained many positive work to the order results of interest results. For instance, he showed 1,092 = PSL2(13) . He that odd-order groups of order | | also proved some of the and importance.” <40,000 were solvable, and earliest theorems on the ditto for odd-order transitive orders of simple groups, permutation groups of degree showing, for instance, that if they are even, they either a prime, or < 100. The fact that some of the must be divisible by 12, 16, or 56. With the aid of proofs involved character theoretic arguments these theoretical results, which he summarized in prompted Burnside to make the following pre- the first edition of [B 1], Burnside was confident that scient comment at the end of the introduction to Hölder’s program could be pushed to at least the [B: 1900c]: order 2,000. He observed with premonition, however, that: “As the limit of the order is increased, The results obtained in this paper, par- such investigations as these rapidly become more tial as they necessarily are, appear to me laborious, as a continually increasing number of to indicate that an answer to the inter- special cases have to be dealt with.” Clearly, esting question as to the existence or stronger general results would be desirable! With nonexistence of simple groups of odd what we can now appreciate as truly uncanny fore- composite order may be arrived at by sight, Burnside finished the first edition of his a further study of the theory of group book with the following closing remark:11 characteristics. No simple group of odd order is at pre-

sent known to exist. An investigation as By the time he published the second edition of [B 1], it was clear that Burnside was morally convinced 10Five years later, Burnside gave a presentation of Cole’s that odd-order groups should be solvable. Short of group by generators and relations, and observed that it making a conjecture, he summarized the situa- is isomorphic to PSL 2(8) in [B: 1898d]. tion by stating (in Note M, p. 503) that: “The con- 11 trast that these results shew between groups of odd Notes to §260 in [B 1, 1st ed., p. 379].

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and even order suggests inevitably that simple Working in the setting of representation theory, groups of odd order do not exist.” Burnside was able to give an affirmative answer to Burnside was not to see a solution of the odd- (2) in the case of complex linear groups. In fact, his order problem in his lifetime; in fact, there was not methods showed that, if G is a subgroup of GLn(C) much progress on the problem to speak of for at for some n, and G has exponent N, then G N3. least another forty-five years. Then, with Brauer’s This result was proved by a trace argument,| |≤ obvi- new idea of studying simple groups via the cen- ously inspired by Burnside’s then ongoing work on tralizers of involutions gradually taking hold, new characters. Burnside also showed that the answer positive results began to emerge on the horizon. to (2) is yes for any group G with exponent N 3. Finally, building on work of M. Suzuki, M. Hall and Later, Schur gave an affirmative answer to (1)≤ for themselves, Feit and Thompson succeeded in prov- any G GLn(C), and Kaplansky extended Schur’s ⊆ ing the solvability of all odd-order groups in 1963. result to G GLn(k) for any field k; details of the Their closely reasoned work [FT] ⊆ proofs can be found in [L, §9]. of 255 pages occupied a single issue of the Pacific Journal of Progress on Burnside’s Prob- Mathematics. Burnside was lems (1) and (2) was at first very proved to be right not only in “The contrast slow. A positive solution for (2) “conjecturing” the theorem, but that these was furnished for N =4by I. N. also in predicting the important Sanov in 1948, and for N =6by role that character theory would results shew M. Hall in 1958. For N 72, P. S. play in its proof. Indeed, Chapter Novikov announced a≥ negative V of the Feit-Thompson paper, al- between answer to (2) in 1959; however, most 60 pages in length, relies al- the details were never published. most totally on working with char- groups of odd Finally, for N odd and 4381, acters and Frobenius groups. Feit the negative answer to≥ (2) ap- and Thompson received the Cole and even peared in the joint work of P. S. Prize for this work in 1965, and Novikov and S. I. Adian in 1968. Thompson was awarded the order suggests For small values of Fields Medal in 1970 for his sub- inevitably that N/ 2, 3, 4, 6 or N even, ap- sequent work on minimal simple parently∈{ not much} is known. In groups. All of this work culmi- simple groups particular, the cases N =5, 8 nated later in the classification seem to be still open. As for Prob- program of finite simple groups of odd order lem (1), the answer turned out to in the early 1980s. be much easier. In 1964, E. S. The spectacular successes of do not exist.” Golod produced for every prime this program have apparently ex- p an infinite group on two gen- ceeded even Burnside’s dreams, erators in which every element for he had stated on page 370 in has order a finite power of p; this the first edition of [B 1] that “a complete solution disposed of Burnside’s Problem (1) in the negative. of this latter problem is not to be expected.” That This was, however, not the end of the story. Since was, however, in the “dark ages” of the 1890s. the 1930s, group theorists have considered an- Burnside would probably have felt very differently other variant of the Burnside Problems, which we a b if he had known the p q theorem, the odd-order can formulate as follows. For given natural num- theorem, and the existence of some of the sporadic bers r and N, let B(r,N) be the “universal Burn- simple groups found in the 1960s and 1970s. side group” with r generators and exponent N; in Today I think it is generally agreed that the clas- other words, B(r,N) is the quotient of the free sification program of finite simple groups could group on r generators by the normal subgroup gen- not have been possible without the pioneering ef- erated by all Nth powers. Burnside’s Problem (2) forts of Burnside. above amounts to asking whether B(r,N) is a fi- Another well-known group-theoretic problem nite group. The following variant of this problem that came from Burnside’s work in 1902–05 con- is called cerns the structure of torsion groups (groups all of whose elements have finite order). There are two The Restricted Burnside Problem. For given nat- (obviously related) versions of this problem, which ural numbers r and N, are there only finitely many may be stated as follows: finite quotients of B(r,N)? Burnside Problem (1). Let G be a finitely gener- The point is that, even if the universal group ated torsion group. Is G necessarily finite? B(r,N) is infinite, one would hope that there are only finitely many ways of “specializing” it into fi- Burnside Problem (2). Let G be a finitely gener- nite quotients (and therefore a unique way to spe- ated group of finite exponent N (that is, gN =1 cialize it into a largest possible finite quotient). In for any g G). Is G necessarily finite? ∈

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1959, A. I. Kostrikin announced a positive solution of B(G) is connected, and there is even a similar to this problem for all prime exponents; much of characterization of minimal simple groups G in his work (and that of the Russian school) is re- terms of B(G). When I saw Louis Solomon in April ported in his subsequent book Around Burnside. 1997 at an MSRI workshop on the interface of rep- After the partial negative solution of the Burnside resentation theory and combinatorics, I asked him Problem (2) was known, the interest in the Re- if the term “Burnside ring” originated with his stricted Burnside Problem intensified. The break- paper [So]. He confirmed this, but added emphat- through came in the early 1990s when E. Zelmanov ically, “It is all in Burnside!” came up with an affirmative solution to this problem, for all r and all N. Surprisingly (to others if A Tale of Two Mathematicians not to experts), Zelmanov’s solution depends heav- As I contemplated and wrote about the career and ily on the methods of Lie algebras and Jordan al- work of F. G. Frobenius and W. Burnside, I could gebras. Another ingredient in Zelmanov’s solution not help noticing the many interesting parallels be- is the classification of finite simple groups: some tween these two brilliant mathematicians. There consequences of the classification theorem were was as much difference in style between them as used in reducing the general exponent case to the one would expect between a German and an Eng- case of prime power exponent via the earlier re- lishman, and yet there were so many remarkable sults of Hall and Higman. Zelmanov’s main work similarities in their mathematical lives that it is was then to affirm the Restricted Burnside Prob- tempting for us to venture a direct comparison. k lem first for N = p with p odd [Z 1], and then for Burnside was three years Frobenius’s junior, k the (much harder) case N =2 [Z 2]. For this work, and survived him by ten, so they were truly con- Zelmanov received the Fields Medal in 1994. Look- temporaries. Coincidentally, they were elected to ing back, I think it is quite remarkable that Burn- the highest learned society of their respective side’s work in representation theory and the open countries in the same year, 1893: Frobenius to the problems he proposed actually spawned the later Prussian Academy of Sciences, and Burnside to work of two Fields Medalists. What a tremendous the Royal Society of England. Mathematically, both legacy to mathematics! started with analysis and found group theory as Some of my teachers and mentors have always the subject of their true love in their mature years. urged me to “read the masters”: they taught me Both got into group theory via the Sylow Theorems, that the great insight of the masters, implicit or and published their own proofs of these theorems explicit in their original writing, is not to be missed for abstract groups: Frobenius in 1887, and Burn- at any cost. In closing this section, I think I’ll pass side in 1894. Other group theory papers of Burn- on this cogent piece of advice to our younger col- side in the period 1893–96 also in part duplicated leagues, using Burnside’s book [B 1] again as a case results obtained earlier by Frobenius. Obviously, in point. There is so much valuable information Frobenius had the priority in all of these, and Burn- packed into this classic that sometimes it is left side felt embarrassed about not having checked the to later generations to unearth the “treasures” that literature sufficiently before he published his own the great master (knowingly or sometimes even un- work. Burnside learned a valuable lesson from this knowingly) left behind. In §§184–185 in the sec- experience, and from that time on, he was to fol- ond edition of [B 1], Burnside discussed the char- low Frobenius’s publications very closely. In his acters of transitive permutation representations of subsequent papers, he made frequent references a group G by making a “table of marks” (their to Frobenius’s work, always referring to him po- character values), and showed how to “compound” litely as “Herr Frobenius” or “Professor Frobenius”. such marks and resolve the results into integral In Burnside’s group theory book [B 1], Frobenius re- combinations of the said marks. More than a half ceived more citations than any other author, in- century later, L. Solomon resurrected this idea in cluding Jordan and Hölder. Frobenius was, how- [So], and formally constructed the commutative ever, less enthused about Burnside’s work, at least Grothendieck ring of the isomorphism classes of at the outset. In his May 7, 1896, letter to Dedekind, finite G-sets, which he appropriately christened the Frobenius wrote,13 after mentioning an 1893 paper “Burnside ring” of the group. Today this Burnside of Burnside on the group determinant: ring B(G) is an important object not only in representation theory, but also in combinatorics and This is the same Herr Burnside who an- topology (especially homotopy theory).12 Some of noyed me several years ago by quickly the connections between B(G) and the group G it- rediscovering all the theorems I had self found by later authors are rather amazing. For published on the theory of groups, in instance, A. Dress [Dr] has shown that G is a solv- the same order and without exception: able group if and only if the Zariski prime spectrum first my proof of Sylow’s Theorems, then the theorem on groups with 12A good reference for this topic is [CR], where the entire square-free orders, on groups of order last chapter is devoted to the study of Burnside rings and 13 their modern analogues, the representation rings. English translation following [H3, p. 242].

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“A lemma that is not Burnside’s” paq, on groups whose order is a prod- 4. Solvability of 1 uct of four or five prime numbers, etc. paqb groups: This #(G-orbits on S)= π(g) G etc. was clearly a com- g G | | X∈ mon goal that both If the above sentiment was expressed in 1896, we Frobenius and where π(g) is the number of points in S can imagine how Frobenius felt later when he saw Burnside had very fixed by g. According to R. C. Read [PR, p. Burnside’s papers [B: 1898a, 1900b], etc., in which much hoped to at- 101], “this lemma has been likened to the Burnside re-derived practically all of Frobenius’s tain. If m is the ex- country yokel’s method of counting cows, results on the group determinant, group charac- ponent of p mod- namely count the legs and divide by four.” ters, and orthogonality relations! At least once or ulo q, Burnside Ironically, it is sometimes easier to count twice (e.g., on page 269 of the second edition of furnished a posi- legs than to count cows! For instance, to count the number of different necklaces a [B 1]), Burnside had stated that in [B: 1898a] he tive solution in had “obtained independently the chief results of case a<2m [B 1 , jeweler can make using six beads of two Professor Frobenius’ earlier memoirs.” For an ex- 1st ed., p. 345], and colors (say green and white), we can get the pert analysis of this claim of Burnside, we refer the Frobenius later re- answer, 13, by applying the above formula reader to [H , p. 278]. laxed Burnside’s to the dihedral group of twelve elements 2 6 It was perhaps a stroke of fate that the group- hypothesis to acting on a set of 2 =64 “formal neck- a 2m. The truth laces”. Burnside deserved credit for popu- theoretic work of Frobenius and Burnside remained ≤ perennially intertwined: they were interested in of the result in all larizing the Cauchy-Frobenius formula by the same problems, and in many cases they strived cases was proved including it in his book. Later, a far-reach- ing generalization of this formula known to get exactly the same results. The following are by Burnside in as Pólya’s Fundamental Theorem became some interesting comparisons. 1904 (Theorem 4.4 a major landmark in the field of enumer- 1. Both Frobenius and Burnside worked on the above); here, Burn- ative combinatorics. question of the existence of normal p-complements side’s acumen with From the viewpoint of representation in finite groups, and each obtained significant con- the arithmetic of theory, π is the character of the permu- ditions for the existence of such complements. characters gave tation representation associated with the Their conditions are different, and the results they him the winning action of G on S. In case this action is obtained are both standard results in finite group edge. doubly transitive, Burnside showed in his theory today. Frobenius’s result seems stronger Since Frobenius book that π is the sum of the trivial char- here, since it gives a necessary and sufficient con- and Burnside acter and an irreducible character of G. dition, while Burnside’s result offers only a suffi- worked on many cient condition. common problems 2. On transitive groups of prime degree: a topic and obtained re- of great interest to Frobenius. Here, Burnside had lated results on them, it is perhaps not surprising the scoop, as he proved in [B: 1900c] that any such that posterity sometimes got confused about which group is either doubly transitive or metacyclic, result is due to which author. One of the most con- from which it follows that there are no simple spicuous examples of this is the famous counting groups of odd (composite) order and prime degree. formula, which says that, with a finite group G act- The paper [B: 1900c] appeared heel-to-heel fol- ing on a finite set S, the average number of fixed lowing Burnside’s paper [B: 1900b] on “group-char- points of the elements of G is given by the num- acteristics”, and represented the first applications ber of orbits of the action (see box). Starting in the of group characters to group theory proper, a fact mid-1960s, more and more authors began to refer acknowledged by Frobenius himself. to this counting formula as “Burnside’s Lemma”. 3. On Frobenius groups: Burnside had been According to P. M. Neumann [Ne], S. Golomb and keenly interested in these groups, and devoted N. G. de Bruijn first made references and attribu- pages 141–144 in [B 1, 1st ed.] and subsequently tions to Burnside for this result in 1961 and [B: 1900a] to their study. He was obviously trying 1963–64, after which the name “Burnside’s Lemma” to prove that the Frobenius kernel is a subgroup, began to take hold. While Burnside did have this and by 1901, he was able to prove this in case the result in his group theory book [B 1, p. 191], he had Frobenius complement has even order or is solv- basically little to do with the lemma. In his paper able. If one assumes the Feit-Thompson Theorem, “A lemma that is not Burnside’s” [Ne], Neumann this would give a de facto proof of the desired con- reported that Cauchy was the first to use the idea clusion in all cases. Maybe this was one of the rea- of the said lemma in the setting of multiply tran- sons that fueled Burnside’s belief that odd order sitive groups, and it was Frobenius who formulated groups are solvable? We do not know for sure. the lemma explicitly in [F: (36), p. 287], and who Anyway, on the Frobenius group problem, it was first understood its importance in applications. Frobenius who had the scoop, as he proved that Neumann’s recommended attribution “Cauchy- the Frobenius kernel is a subgroup in all cases in Frobenius Lemma” was lauded by de Bruijn in a 1901. Frobenius’s great expertise with induced quotation at the end of [Ne]; however, in his group characters gave him the edge in this race. theory book [NST] with Stoy and Thompson, Neu-

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mann somehow decided to refer to the result as [H2] T. HAWKINS, Hypercomplex numbers, Lie groups, and “Not Burnside’s Lemma”! the creation of group representation theory, Archive Another case in point is the theorem, mentioned Hist. Exact Sci. 8 (1971), 243–287. already in Part I, that the degree of an irreducible [H3] ———, New light on Frobenius’ creation of the theory of group characters, Archive Hist. Exact Sci. 12 (complex) representation of a group G divides the (1974), 217–243. order of G. Some authors have attributed this the- [L] T. Y. LAM, A first course in noncommutative rings, orem to Burnside, but again it was Frobenius who Graduate Texts in Math., vol. 131, Springer-Verlag, first proved this result, as one can readily check Berlin, 1991. a b by reading the last page of his classical group de- [Mat]H. MATSUYAMA, Solvability of groups of order 2 p , terminant paper [F: (54)].14 Burnside supplied a Osaka J. Math. 10 (1973), 375–378. proof of this result in his own terms, but the the- [M1] T. MOLIEN, Über Systeme höherer complexer Zahlen, orem was definitely Frobenius’s. Issai Schur, a stu- Math. Ann. 41 (1893), 83–156; Berichtigung: Math. dent of Frobenius, proved later that the degree of Ann. 42 (1893), 308–312. an irreducible representation divides the index of [Ne] P. M. NEUMANN, A lemma that is not Burnside’s, Math. Scientist 4 (1979), 133–141. the center of G, and N. Itô was to prove eventu- [NST]P. M. NEUMANN, G. STOY, and E. THOMPSON, Groups ally that this degree in fact divides the index of any and geometry, Oxford Univ. Press, 1994. abelian normal subgroup. [N] E. NOETHER, Hyperkomplexe Grössen und Darstel- With such little tales on attributions, we con- lungstheorie, Math. Z. 30 (1929), 641–692. clude our discussion of the life and work of Frobe- [PR] G. PÓLYA and R.C. READ, Combinatorial enumeration nius and Burnside. Although their work was so of groups, graphs, and chemical compounds, closely linked, there seemed to have been no evi- Springer-Verlag, Berlin, 1987. dence that they had either met, or even corre- [So] L. SOLOMON, The Burnside algebra of a finite group, sponded with each other. Would the history of the J. Combin. Theory 2 (1967), 603–615. representation theory of finite groups be any dif- [Z1] E. ZELMANOV, Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR-Izv. 36 ferent if these two great mathematicians had (1991), 41–60. known each other, or if there had been a Briefwech- [Z2] ———, A solution of the restricted Burnside problem sel between them, like that between Frobenius and for 2-groups, Math. USSR-Sb. 72 (1992), 543–564. Dedekind?

14The result was also proved independently by Molien, as pointed out by Hawkins [H 2, p. 271].

References [Ab] S. ABHYANKAR, Galois theory on the line in nonzero characteristic, Bull. Amer. Math. Soc. (N. S.) 27 (1992), 68–133. [Be] H. BENDER, A group-theoretic proof of Burnside’s paqb-theorem, Math. Z. 126 (1972), 327–328. [B] W. BURNSIDE, Bibliography (compiled by A. Wagner and V. Mosenthal), Historia Math. 5 (1978), 307–312. [B 1] ———, Theory of groups of finite order, Cambridge, 1897; second ed., 1911, reprinted by Dover, 1955. [Co] F. N. COLE, Simple groups as far as order 660, Amer. J. Math. 15 (1893), 303–315. [Cu2]C. W. CURTIS, Frobenius, Burnside, Schur and Brauer: pioneers of representation theory, Amer. Math. Soc., Providence, RI, to appear. [CR] C. W. CURTIS and I. REINER, Methods of representation theory, Vol. 2, Wiley, New York, 1987. [Dr] A. W. M. DRESS, A characterization of solvable groups, Math. Z. 110 (1969), 213–217. [FT] W. FEIT and J. G. THOMPSON, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. [Fo] A. R. FORSYTH, William Burnside, J. London Math. Soc. 3 (1928), 64–80. (Reprinted in Burnside’s Theory of Probability, Cambridge Univ. Press, 1928.) [F] F. G. FROBENIUS, Gesammelte Abhandlungen I, II, III (J.-P. Serre, ed.), Springer-Verlag, Berlin, 1968. [Go] D. M. GOLDSCHMIDT, A group theoretic proof of the paqb theorem for odd primes, Math. Z. 113 (1970), 373–375. [HR] I. HALPERIN and P. ROSENTHAL, Invariant subspaces, Springer-Verlag, Berlin, 1973.

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