## Traité des Substitutions et des Équations algébriques.

Paris: Gauthier-Villars, 1870.

First edition, very rare, of “the book that established group theory as a subject in its own right in mathematics” (Gray, p. 149). “Jordan’s monumental work, *Traité des Substitutions et des Équations algébriques*, published in 1870, is a masterpiece of mathematical architecture. The beauty of the edifice erected by Jordan is admirable” (Van der Waerden, *A History of Algebra*, p. 117). “In 1870, Jordan gathered all his results on permutation groups for the previous ten years in a huge volume, *Traité des Substitutions*, which for thirty years was to remain the bible of all specialists in group theory. His fame had spread beyond France, and foreign students were eager to attend his lectures; in particular Felix Klein and Sophus Lie came to Paris in 1870 to study with Jordan” (DSB). “An instant classic, his *Traité *set a new research agenda of creating a *theory* of groups as opposed to the older agenda of devising ways to *calculate* the solutions of polynomial equations” (Katz & Parshall, p. 316). “The title of this comprehensive work of 667 quarto pages is excessively modest and therefore misleading. The work represents not only the definitive solution of the problem formulated by Galois, but also a review of the whole of contemporary mathematics from the standpoint of group-theoretic thinking” (Wussing, pp. 141-142). “Jordan’s place in the tradition of French mathematics is exactly halfway between Hermite and Poincaré. Like them he was a ‘universal’ mathematician who published papers in practically all branches of the mathematics of his time … [but] it is chiefly as an algebraist that he reached celebrity when he was barely thirty; and during the next forty years he was universally regarded as the undisputed master of group theory. When Jordan started his mathematical career, Galois’s profound ideas and results (which had remained unknown to most mathematicians until 1846) were still very poorly understood, despite the efforts of Serret and Liouville to popularize them; and before 1860 Kronecker was probably the only first-rate mathematician who realized the power of these ideas and who succeeded in using them in his own algebraic research. Jordan was the first to embark on a systematic development of the theory of finite groups and of its applications in the directions opened by Galois … He also was the first to investigate the structure of the general linear group and of the ‘classical’ groups over a prime finite field, and he very ingeniously applied his results to a great range of problems; in particular, he was able to determine the structure of the Galois group of equations having as roots the parameters of some well-known geometric configurations (the twenty-seven lines on a cubic surface, the twenty-eight double tangents to a quartic, the sixteen double points of a Kummer surface, and so on)” (*ibid*.). This is an extremely rare book on the market, and very uncommon even in institutional collections. ABPC/RBH list only the two copies in the Duarte sale in 1977. It has been suggested that most copies of the book were destroyed in a fire at the publisher’s warehouse during the violent suppression of the Paris Commune early in 1871, and that the marginal browning seen in many of the surviving copies was caused by the heat of the fire.

Évariste Galois (1811-32) published a few short papers in his lifetime, but his most important works were posthumous. He first set down his ideas on the relationship between what we call group theory and the solvability of polynomial equations in his most important work, ‘Mémoire sur les conditions de résolubilité des équations par radicaux,’ usually called the ‘Premier Mémoire,’ which he submitted to the Paris Académie des Sciences but which they rejected and returned to the author on 4 July 1831. He followed this with ‘Des équations primitives qui sont solubles par radicaux,’ also known as the *Second Mémoire*. Both *M**émoires *were published for the first time, together with Galois’s other works, by Joseph Liouville in his *Journal de Mathématiques pures et appliquées *in 1846, and it was through this publication that Galois’s works became known to the wider mathematical world.

“The publication of Galois’s work in Liouville’s *Journal* was a challenge to all mathematicians to understand it, extend it, and apply it. Ultimately, it stimulated the emerging generation of mathematicians, as Wussing has described. He noted an initial period in which Betti, Kronecker, Cayley, Serret, and some others filled in holes in Galois’s presentation of the idea of a group. These modest yet difficult pieces of work established the connection between group theory and the solvability of equations by radicals [i.e., by expressions involving the sums, differences, products and quotients of whole numbers and their square, cube and higher roots], and then explored the solution of equations by other means than radicals. The implicit idea of a group was expressed in terms of permutations of a finite set of objects, amalgamating Cauchy’s presentation of the theory of permutation groups in 1844-46 and Galois’s terminology...

“The crucial presentations of the idea of permutation groups were made by Jordan in his ‘Commentaire sur Galois’ [*Mathematische Annalen* 1 (1869), 141-160] and his *Traité des Substitutions et des Équations algébriques *(1870). Jordan’s systematic theory of permutation groups was much more abstract; he spoke of abstract properties such as commutativity, conjugacy, centralizers, transitivity, ‘normal’ subgroups (and, one might say obliquely, of quotient groups), group homomorphisms and isomorphisms. So much so that one can argue that Jordan came close to possessing the idea of an abstract group. Jordan said (*Traité*, p. 22), ‘One will say that a system of substitutions form a group if the product of two arbitrary substitutions of the system belongs to the system itself.’ He spoke (*Traité*, p. 56) of isomorphisms (which he called an *isomorphisme holoédrique*) between groups as one-to-one correspondences between substitutions which respect products...

“A further indication of the high level of abstraction at which Jordan worked is his use of technical concepts of increasing power … Another is the wide range of situations in the *Traité* in which groups could be found permuting geometrical objects: the 27 lines on a cubic surface, the 28 bitangents to a quartic, the symmetry groups of the configuration of the nine inflection points on a cubic, and of Kummer’s quartic with sixteen nodal points. Powerful abstract theory and a skilled recognition of groups ‘in nature’ suggests that Jordan had an implicit understanding of the group idea that he presented in the language of permutation groups only for the convenience of his audience. This is not to deny the role of permutation-theoretic ideas in Jordan’s work, indicated by the emphasis on transitivity and degree (= the number of elements in the set being permuted), but rather to indicate that ideas of composition and action … were prominent and could be seized upon by other mathematicians” (Gray, pp. 152-153).

“The *Traité des Substitutions et des Équations algébriques* begins with an introduction of extraordinary scope, which demonstrates the scientific staring point, as well as the greatness and limitations, of Jordan’s conception. In a certain sense the *Traité* contains formulations of problems belonging to a development yet to come …

“Jordan begins his introduction by stressing the fundamental difference between Galois’s papers on the theory of equations and the relevant papers of Galois’s predecessors from Lagrange to Abel. By associating to every equation a (permutation) group whose structure mirrors its essential properties, including its possible solution by radicals, Galois had supplied the ultimate basis of the theory of equations. This assigned a special role to the theory of permutations. As the foundation of all questions bearing on the theory of equations, permutation theory became an independent area of investigation … In view of this development and of algebraic questions that had just arisen, Jordan concludes that the theory of equations had acquired a new and very different character. The old quest for solutions of given equations in the form of transparent solution formulas had become the study of the structure of algebraic number fields [although Jordan did not use the concept of a field] …

“He designates in the introduction two large groups of problems to be studied in the *Traité* with the aid of permutation theory:

- The study of ‘division’ of transcendental functions [expressing
*f*(*x/n*) in terms of*f*when*f*is an elliptic function]. This study had already given Galois the opportunity to make a new and brilliant application of his method. It had been advanced by Hermite who, using Galois’s methods, built on the relevant work of Abel and Gauss … - The use of permutation theory in studying the new direction of developments of analytic geometry. This new development had been initiated by O. Hesse, largely in the fifties, in papers on the number of inflection points of cubic curves. Through these papers, algebra, with its then very modern tools, began to appear as the direct representation of geometric propositions …

“It thus appears that Jordan had grasped an objective tendency that pointed to the use of group theory in geometry and that was to gain general acceptance two years after the *Traité* as a result of [Klein’s] Erlangen Program … He was able to classify earlier, isolated results by group-theoretic means; more specifically, he could describe such results in terms of statements about subgroups of the ‘linear group’, a group that he had investigated in great detail” (Wussing, pp. 154-156).

“The short Livre I picks up on the topic of Galois’s ‘Sur la théorie des nombres’ [*Bulletin des Sciences Mathématiques, Physiques et Chimiques de M. Férussac* 13 (1830), 428-435] and is about congruences modulo a prime or prime power (what we today would call finite fields). The material on modular arithmetic is needed because the *Traité* is entirely about finite groups (often described as matrix groups with entries modulo a prime number).

“Livre II is about substitutions (Galois’s word for what would later become permutations). It ranges widely, bringing in work by Lagrange, Cauchy, Mathieu, Kirkman, Bertrand, and Serret. Its themes are transitivity, simple and multiple, primitivity (the topic of Galois’s second memoir) and composition factors. It also gives a (flawed) proof that the alternating group *A _{n}* is simple when

*n*≠ 4 [the alternating group consists of permutations that can be effected by making an even number of exchanges of pairs of elements]. Here Jordan presented the opening propositions of the theory of groups: Lagrange’s theorem and Cauchy’s theorem. Lagrange’s theorem [and proof] is stated in the form it retains to this day: the order of [= number of elements in] a subgroup divides the order of the group … Cauchy’s theorem (§40) is the partial converse: if a prime

*p*divides the order of a group then there is an element of order

*p*in the group [i.e., an element whose

*p*th power is the identity element] …

“Then comes an account of transitivity, and the ‘orbit-stabilizer’ theorem: Jordan proved (§44) that given a set of objects permuted by a group *G*, if these elements can be sent to *m* different systems of places, and *n* is the order of the subgroup that leaves these elements fixed, then the order of the group is *mn*. Jordan was very interested in how transitive a group could be, and in §47 recorded the ‘remarkable’ example of the Mathieu group on 12 letters that is 5-fold transitive [i.e., any 5 different letters can be sent to any other (or the same) 5 different letters by some element of the group].

“Then we get some 160 pages on what we might call finite linear groups, groups whose elements are matrices with entries in a finite field. They come in various types, and the names have not always retained the meaning Jordan gave them: primary, orthogonal, abelian, hyperabelian. A typical theme is the ‘normal’ subgroups and composition factors of the different groups. He was also interested in the concept of primitivity, which he defined negatively: a group is non-primitive (better, imprimitive) if the elements [being permuted by the group] can be divided into blocks containing the same number of elements and the group maps blocks to blocks.

Livre III, ‘The irrationals,’ is about the behaviour of a given irreducible equation under successive adjunction of irrationals (roots of other polynomial equations). It deals with the solution of the quartic by radicals and the insolubility of the general quintic. Then we get a treatment of equations with particular kinds of group: abelian groups and what Jordan called ‘Galois’s equation’ *x ^{p} = A*. Then we get the geometrical examples mentioned above in which discoveries by Hesse, Clebsch, Kummer and Cayley (the 27 lines on a cubic surface) are shown to have interesting group-theoretic interpretations. Then comes material from elliptic function theory: the modular equation and the discovery in 1858 of the solution of the quintic equation according to Hermite and Kronecker. Jordan reworked their contributions in his own way and then extended their work to show that all polynomial equations can be solved by a similar use of hyper-elliptic functions.

“The books ends with Livre IV, of almost 300 pages, entitled ‘Solution by radicals.’ Jordan quickly rehearsed the argument that an equation is solvable by radicals if and only if the composition factors of its group are all prime, and called such a group solvable. This led him to proclaim three problems, of which for brevity I give the first only: construct explicitly, for every degree the general [i.e., maximal] solvable transitive groups … Jordan used his theory of composition factors to suggest that a massive process of induction might suffice to find all permutation groups … it might be possible to construct the given group from its composition series if all groups of order less than the given group are known. In the event this ambitious programme failed – groups are vastly more varied than can be handled this way – but it has the germ of a process that was later made more precise by Hölder and became the search for all simple groups” (Gray, pp. pp. 153-154).

Some, possibly all, of the four ‘Livres’ into which the *Traité** is divided may have been issued separately: we have seen copies of two or three of the Livres bound separately in contemporary bindings.*

Marie-Ennemond-Camille Jordan (1838-1922) was a professor of mathematics at the École Polytechnique in Paris from 1876 to 1912. He edited Liouville’s *Journal des mathématiques pures et appliquées* from 1885 until his death. Apart from the *Traité des Substitutions**, *which brought him the Poncelet Prize of the French Academy of Sciences, he published his lectures and researches on analysis in the influential *Cours d’analyse de l’École Polytechnique* (three vols., 1882-87), about which the great English mathematician G. H. Hardy wrote: “[As a student I was advised] to read Jordan’s *Cours d’analyse*; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.” In the third edition of this work (1909-15), Jordan gave the proof of what is now known as Jordan’s curve theorem: any closed curve that does not cross itself divides the plane into exactly two regions, one inside the curve and one outside.

Gray, *A History of Abstract Algebra*, 2018. Katz & Parshall, *Taming the Unknown*, 2014. Wussing, *The Genesis of the Abstract Group Concept*, 2007.

4to (269 x 221 mm), pp. xviii, 667 (some marginal browning). Contemporary half-morocco and marbled boards, spine lettered in gilt (spine faded).

Item #4912

**
Price:
$14,500.00
**