Not all that long ago the community of algebraists was all-a-twitter with the completion of the classification of the finite simple groups, these being to finite groups what primes are to the multiplicative structure of the integers, at least after a fashion. Jordan-Hölder is after all a bit trickier than the fundamental theorem of arithmetic. Nonetheless, this part of the structure theory of finite groups is part of even the elementary courses on group theory and belongs to every mathematician’s toolkit, even if it’s somewhere at the bottom for most non-group theorists.
Culturally speaking, however, this classification problem, or project, famously directed and spearheaded by Daniel Gorenstein, and involving, e.g. Michael Aschbacher, Walter Feit, John Thompson, Graham Higman, and John Horton Conway, was certainly on every one’s radar screen in the last decades of the previous century. There are also still a lot of aftershocks, what with Gorenstein’s original program being officially finished only ca. 2004 (at least according to what the ever-more-toxic and ever-more-addictive Wikipedia has to say), one notable theme being the “monstrous moonshine” of Conway, concerning the “Monster” of Griess and Fischer. This moonshine business was settled by Richard Borcherds in 1992.
Thus, whether it be for strictly algebraic reasons, for more ecumenical ones, or simply because of their beauty, simple groups are very important and nigh-on irresistible mathematical players. The book under review is the chronicle of the travels of one of the frontier-men in the field: William Burnside himself. This is in fact the second edition (1911), of Burnside’s classic, its first appearance coming in 1897, and constitutes both a major historical event in mathematical publication and a marvelous, if dated, course in group theory competent to take the reader from novice to devotee in one orbit (if I may be forgiven this choice of word).
There is an obvious caveat in that the style and language (and some of the notation) employed by Burnside is to our tastes a little archaic. It should be noted that Burnside does not use the “theorem-proof” template that is all-but-ubiquitous these days, but this by no means detracts from the rigor and even the clarity of the presentation. Indeed, there is a gain in clarity to be had, if one properly immerses oneself in the narrative. Consider, e.g., the following random sample (p. 278):
… Hence: — Theorem IV. If H is a self-conjugate [i.e. normal] subgroup of G, and if G/H has r' sets of conjugate operations, there are at least r' distinct irreducible representations of G, in each of which the identical substitution [i.e. the identity mapping] corresponds to every operation of H. The converse of this theorem will be considered in the following chapter …, and it will be seen that r' is the actual number of representations of G which have the property in question …
What this illustrates is that Burnside’s treatment of what would today likely be presented with a lot more machinery in play is much more along the lines of a “nuts-and-bolts” approach, which is of course hugely beneficial to any one wishing to learn the craft of finite group theory as, for lack of a better word, a combinatorial affair. And this is clearly a great virtue.
Theory of Groups of Finite Order is a big book (over 500 pages), encyclopedic (modulo its period) even as it starts with the very basics of its subject, and presents major results at the hand of a true master of group theory. Not only are Burnside’s own researches well-represented, with his famous result that “[a] group whose order contains only two distinct primes is soluble” occurring on p. 323, but his treatment of, for example, Sylow’s theorems in Chapter IX (p. 149, ff.) is both gorgeous and extremely readable. This is the rule, not the exception. To boot, there are plenty of exercises strewn throughout the book, testifying to Burnside’s pedagogical objectives in writing this opus. Abel was right: what better way can there be to learn the craft of doing finite group theory than to work carefully through the pages of a masterpiece like this?
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.