Camille Jordan

How to use this course

Every mathematics student meets Camille Jordan several times without learning who he was. Linear algebra has the Jordan normal form, group theory the Jordan–Hölder theorem, analysis the Jordan decomposition of a function of bounded variation and Jordan content, topology the Jordan curve theorem. For forty years he was the leading authority on finite groups, and his Traité des substitutions of 1870 turned Évariste Galois’s ideas into a working theory.

He did most of this research while employed as a mining engineer. This course follows his life in Parts I and III, and his mathematics, area by area, in Part II.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–07 cover the algebra (1860–1880); Lessons 08–10 cover the analysis and topology of the Cours d’analyse (1881–1893).

Timeline

Lyon and the Polytechnique, 1838–1861

Marie Ennemond Camille Jordan was born on 5 January 1838 in La Croix-Rousse, the silk-weavers’ quarter on the hill above Lyon. The family was prosperous and well connected. His father, Esprit-Alexandre Jordan, was an engineer who had studied at the École Polytechnique. His mother, Joséphine Puvis de Chavannes, was the sister of Pierre Puvis de Chavannes, later one of the most celebrated French painters of the century. A great-uncle, also named Camille Jordan, had been a politician during the Revolution and the Restoration, and a cousin, Alexis Jordan, was a botanist.

Jordan attended the Lycée de Lyon and the Collège d’Oullins. In 1855, aged seventeen, he entered the École Polytechnique, and on graduating he joined the state corps of mining engineers. The path was the standard one for a gifted French mathematician of his class, and it gave him a secure salary for life.

On 14 January 1861 he defended two doctoral theses in Paris before Jean-Marie Duhamel, Joseph-Alfred Serret and Victor Puiseux. The first, on algebra, concerned the number of distinct values a function takes when its variables are permuted, the problem that Joseph-Louis Lagrange and Augustin-Louis Cauchy had made central to the theory of equations. The second, proposed by Puiseux, concerned the periods of inverse functions of integrals of algebraic differentials. The Mathematics Genealogy Project lists Puiseux and Serret as his advisors.

The engineer who read Galois, 1861–1870

For the next twelve years Jordan worked as an engineer, first at Privas in the Ardèche, then at Chalon-sur-Saône, and finally in Paris. The posts left him time for research, and almost all of his roughly 120 papers were written before he left the engineering corps. He married Marie-Isabelle Munet in 1862. They had two daughters and six sons.

His great subject was the work of Évariste Galois, which Joseph Liouville had published in 1846. By the 1860s few mathematicians understood it. Serret’s algebra textbook included Galois theory only in its third edition of 1866. Jordan set out to master it completely: he published a commentary on Galois’s memoir in 1865 and a long series of papers on groups of permutations, which he called substitutions, and their application to equations.

He worked in other fields too. In 1866 he published papers on the deformation of surfaces and on closed curves drawn on surfaces. In them he studied when one closed path on a surface can be continuously deformed into another, the idea now called homotopy, and attempted a classification of surfaces up to deformation. In 1868–69 he published a memoir on groups of motions of space (Lesson 07).

The Traité and the summer of 1870

In 1870 Jordan gathered a decade of results into the Traité des substitutions et des équations algébriques, a book of almost 700 pages. It is often called the first book on group theory. It contains the theory of permutation groups, composition series, the linear groups over the integers modulo a prime with their canonical form, solvable groups, and applications of Galois theory to classical problems: the solution of equations by radicals, the 27 lines on a cubic surface, the bitangents of a quartic curve, and the division of elliptic and hyperelliptic functions. The Academy awarded it the Poncelet Prize. For about thirty years it was the standard reference for every specialist in the subject.

The book drew two young foreign mathematicians to Paris. Sophus Lie and Felix Klein, who had met in Berlin, arrived in the spring of 1870 and studied with Jordan. They absorbed his view that groups should be central to geometry. The visit ended abruptly when the Franco-Prussian War broke out in July. Klein, a Prussian, left at once. Lie set out on foot toward Italy and was arrested near Fontainebleau as a suspected spy, his notes on line complexes taken for coded messages; Gaston Darboux secured his release a month later.

The two visitors went on to create the theory of continuous groups (Lie) and the Erlangen programme of 1872, which classifies geometries by their groups of transformations (Klein). Historians have suggested that Jordan’s memoir on groups of motions was one of their sources.

Composition series and Jordan–Hölder

For the general reader. Every whole number breaks down into primes in only one way. Jordan found that finite groups have a similar property. A group can be broken down step by step into simple pieces, groups that cannot be broken further. There may be many ways to do the breaking, but the pieces that come out are always the same. Jordan proved half of this in 1869, and Otto Hölder completed it in 1889.

Definitions

A composition series of a finite group \(G\) is a chain

in which each \(G_{i-1}\) is normal in \(G_i\) and each quotient \(G_i/G_{i-1}\) is simple and non-trivial. The quotients are the composition factors. Every finite group has a composition series: refine any chain by inserting a maximal normal subgroup at each step.

Jordan proved that the orders \(|G_i/G_{i-1}|\) are the same up to order. Hölder, who had the concept of the quotient group that Jordan lacked, proved that the factors themselves are isomorphic.

Proof

Induct on \(|G|\). Let \(1=G_0\trianglelefteq\cdots\trianglelefteq G_r=G\) and \(1=H_0\trianglelefteq\cdots\trianglelefteq H_s=G\) be two composition series. If \(G_{r-1}=H_{s-1}\), apply induction to that subgroup. Otherwise let \(N=G_{r-1}\), \(M=H_{s-1}\). Both are maximal normal subgroups, so \(NM=G\), and \(K=N\cap M\) is normal in both. By the second isomorphism theorem,

Take a composition series of \(K\). Then \(N\) has two composition series: the given one, and the one through \(K\), with top factor \(N/K\cong G/M\). By induction they have the same factors. Doing the same for \(M\) and comparing the two series of \(G\) through \(K\), which differ only by swapping \(G/N\) and \(G/M\), completes the proof.

Solvability

A finite group is solvable if all its composition factors are cyclic of prime order. Galois’s theorem, in Jordan’s formulation, says that a polynomial equation over \(\mathbb Q\) is solvable by radicals exactly when its Galois group is solvable. The alternating group \(A_n\) is simple for \(n\ge5\), so \(S_n\) has composition factors \(A_n\) and \(\mathbb Z/2\), and the general equation of degree \(n\ge5\) is not solvable by radicals.

Linear groups and the canonical form

For the general reader. A square matrix can look complicated in one coordinate system and simple in another. Jordan showed in 1870 how to choose coordinates in which any matrix becomes almost diagonal, with its eigenvalues on the diagonal and a few 1s just above. He found this form while studying groups of matrices with entries in the integers modulo a prime, and it is now the standard end point of a linear algebra course.

The canonical form

A Jordan block of size \(k\) with eigenvalue \(\lambda\) is

Jordan stated the theorem for matrices over the field \(\mathbb F_p\), where he called it the forme canonique. Karl Weierstrass had obtained an equivalent classification over \(\mathbb C\) in 1868 with his theory of elementary divisors. In 1874 Jordan and Kronecker argued in print over priority and over whether a canonical form or a set of invariants was the right answer, a dispute studied in detail by the historian Frédéric Brechenmacher.

Proof sketch

The space splits as \(F^n=\bigoplus_\lambda\ker(A-\lambda)^{n}\), the generalized eigenspaces, by the Chinese remainder theorem applied to the factors of the characteristic polynomial. On each piece \(N=A-\lambda\) is nilpotent. For a nilpotent \(N\), choose vectors \(v\) with \(N^{k}v=0\), \(N^{k-1}v\ne0\) whose chains \(v,Nv,\dots,N^{k-1}v\) together form a basis; such a choice exists by induction on \(\dim\) using \(\ker N\subseteq\operatorname{im}N+\ker N\). In the basis \(N^{k-1}v,\dots,Nv,v\), \(N\) acts as \(J_k(0)\).

Linear groups over finite fields

The group \(\mathrm{GL}_n(\mathbb F_p)\) of invertible \(n\times n\) matrices over \(\mathbb F_p\) has order

since the \(i\)-th column can be any vector outside the span of the previous ones. Jordan studied this group and its subgroups, including the symplectic groups, which he called abelian groups after Abel’s work on the division of periods. He showed that the projective special linear groups \(\mathrm{PSL}_n(\mathbb F_p)\) are simple apart from small exceptions, giving the first infinite families of simple groups beyond the alternating groups. Leonard Dickson extended these results to all finite fields in 1901.

Bilinear forms (1874)

Eugenio Beltrami in 1873 and Jordan in 1874 independently showed that a real square matrix \(A\) can be written \(A=U\Sigma V^{\mathsf T}\) with \(U,V\) orthogonal and \(\Sigma\) diagonal with non-negative entries, the singular values, and that these are a complete set of invariants under independent orthogonal changes of variable on each side. This is the singular value decomposition.

Permutation groups and finiteness

For the general reader. A permutation group shuffles a finite set of objects. Most such groups are rather small, but a few, the full symmetric and alternating groups, contain every shuffle or every even shuffle. Jordan proved that a group which shuffles the objects thoroughly and contains certain special shuffles must be one of those giants. He also proved a simple counting fact that turned out to have deep consequences in number theory.

A group \(G\le S_n\) is transitive if it can move any point to any other. It is primitive if it is transitive and preserves no partition of \(\{1,\dots,n\}\) into blocks other than the trivial ones. It is \(k\)-transitive if it can send any ordered \(k\)-tuple of distinct points to any other.

Proof. By the orbit-counting lemma, \(\frac1{|G|}\sum_{g\in G}|\mathrm{Fix}(g)|\) equals the number of orbits, which is 1. The identity fixes \(|X|\ge2\) points, more than the average. So some \(g\) fixes fewer than one point, that is, none.

Jean-Pierre Serre devoted a 2003 survey to this theorem. Applied to the Galois group of an irreducible polynomial \(f\in\mathbb Z[x]\) of degree at least 2 acting on its roots, it implies through the Chebotarev density theorem that \(f\) has no root modulo \(p\) for infinitely many primes \(p\), indeed for a set of primes of positive density.

These results reflect Jordan’s main programme after 1870: to show that primitive groups other than \(A_n\) and \(S_n\) are severely restricted. He proved that the degree of a primitive group not containing \(A_n\) is bounded in terms of its class, the minimal number of points moved by a non-identity element, and he studied how high the transitivity of such groups can be. The answer, that only the Mathieu groups \(M_{11}, M_{12}, M_{23}, M_{24}\) are 4- or 5-transitive apart from \(A_n\) and \(S_n\), required the classification of finite simple groups.

Groups of motions and finite linear groups

For the general reader. A crystal repeats a pattern of atoms in three directions. The symmetries of such a pattern form a group of rigid motions of space. In 1868–69 Jordan set out to list all groups of motions, and his memoir started the road that led to the classification of the 230 crystal symmetry groups. Ten years later he proved that every finite group of matrices is, up to a bounded amount, commutative.

Groups of motions (1868–69)

In his Mémoire sur les groupes de mouvements, published in the Annali di Matematica, Jordan classified the groups of orientation-preserving rigid motions of space that are closed in a suitable sense, including those mixing continuous and discrete parts. His list had gaps and the decisive classification of the 230 space groups came later, independently from Evgraf Fedorov and Arthur Schoenflies in 1891, but Jordan was the first to pose the problem as a question about groups. A basic constraint is the following.

Proof. In a basis of \(\Lambda\) the rotation has an integer matrix, so its trace is an integer. In a rotation basis the trace is \(2\cos(2\pi/n)\) in the plane, or \(1+2\cos(2\pi/n)\) in space. So \(2\cos(2\pi/n)\in\{-2,-1,0,1,2\}\), giving \(n=2,3,4,6,1\).

Finite linear groups (1878)

Lazarus Fuchs had asked when a linear differential equation has only algebraic solutions. Jordan reduced the question to the finite subgroups of \(\mathrm{GL}_n(\mathbb C)\), since the monodromy group of such an equation must be finite, and proved a structure theorem for them.

An abelian finite subgroup of \(\mathrm{GL}_n(\mathbb C)\) can be diagonalized simultaneously, so the theorem says that a finite linear group is diagonal up to a bounded index. Issai Schur extended it to periodic groups, and Michael Collins found the optimal bound \(f(n)=(n+1)!\) for \(n\ge71\) using the classification of finite simple groups; the symmetric group \(S_{n+1}\) acting on its \(n\)-dimensional representation shows this is sharp.

Bounded variation and Fourier series

For the general reader. A function has bounded variation if the total amount it rises and falls is finite. Jordan introduced the idea in 1881 to settle when a Fourier series converges, and showed that such a function is just the difference of two functions that only go up. The concept became fundamental in measure theory, probability and the calculus of variations.

For \(f:[a,b]\to\mathbb R\), the total variation is

and \(f\) is of bounded variation if \(V_a^b(f)<\infty\).

Proof. Monotone functions have variation \(g(b)-g(a)\), and variation is subadditive, so \(g-h\) has bounded variation. Conversely, \(V_a^x\) is non-decreasing, and for \(x<y\), \(h(y)-h(x)=V_x^y(f)-(f(y)-f(x))\ge0\), using the additivity \(V_a^y=V_a^x+V_x^y\).

Peter Gustav Lejeune Dirichlet had proved convergence in 1829 for piecewise monotone functions with finitely many discontinuities. Jordan’s class is exactly the linear span of monotone functions, so the decomposition theorem reduces his result to Dirichlet’s argument. The proof uses the second mean value theorem for integrals with the Dirichlet kernel \(D_N(t)=\frac{\sin((N+\frac12)t)}{\sin(t/2)}\).

Jordan used the same concept to define the length of a curve: a parametrized curve \(t\mapsto(x(t),y(t))\) is rectifiable, with finite length, exactly when \(x\) and \(y\) are of bounded variation.

Content, length and measure

For the general reader. What is the area of a strange shape? Jordan’s answer: cover it from outside and fill it from inside with little squares. If the two estimates agree as the squares shrink, that common value is the area. This is the natural notion behind the Riemann integral. Its limits led Henri Lebesgue, who learned from Jordan’s Cours, to the modern theory of measure.

Let \(E\subset\mathbb R^d\) be bounded. For the grid of cubes of side \(2^{-k}\), let \(\underline{c}_k(E)\) be the total volume of the cubes contained in \(E\) and \(\overline{c}_k(E)\) the volume of the cubes meeting \(\overline E\). The inner and outer content are \(\underline c(E)=\lim\underline c_k\) and \(\overline c(E)=\lim\overline c_k\). Giuseppe Peano introduced these in 1887, and Jordan independently soon afterwards; they appear in the second edition of his Cours (1893).

Proof of the first statement. \(\overline c_k(E)-\underline c_k(E)\) is the volume of cubes that meet \(\overline E\) but are not inside \(E\). Each such cube meets \(\partial E\), and conversely every cube meeting \(\partial E\) is counted, up to its neighbours. So the gap tends to 0 exactly when \(\overline c(\partial E)=0\). Since \(\partial E\) is compact, outer content 0 coincides with Lebesgue measure 0 by a finite subcover argument.

Jordan content is finitely additive but not countably additive. Lebesgue’s thesis of 1902 replaced finite covers by countable ones, and the resulting measure measures far more sets.

The Jordan curve theorem

For the general reader. Draw a closed loop on paper without lifting the pen or crossing your own line. The loop separates the paper into an inside and an outside. This looks too obvious to need proof, but for a loop as wild as a fractal the meaning of “inside” is far from clear. Jordan gave the first proof in 1887. For a century his proof was believed to be flawed; recent scholarship has defended it.

Polygons

Let \(\gamma\) be a simple closed polygon. For a point \(x\notin\gamma\), choose a ray from \(x\) that avoids the vertices, and let \(\pi(x)\in\{0,1\}\) be the parity of the number of times it crosses \(\gamma\). Rotating the ray past a vertex changes the count by 0 or 2, so \(\pi(x)\) does not depend on the ray. Moving \(x\) along a path that avoids \(\gamma\) does not change \(\pi\). Points far away have \(\pi=0\), points just to either side of an edge have different parities, so there are at least two components. There are at most two, because any point can be joined, by a path running parallel to \(\gamma\), to a point near a fixed edge, on one side or the other. The general theorem approximates \(\gamma\) by polygons, and the difficulty lies in controlling the approximation.

History

Bernard Bolzano had recognized that the statement needed proof. Jordan’s proof appeared in volume 3 of the Cours d’analyse in 1887. In 1905 Oswald Veblen published a proof that he presented as the first rigorous one, criticizing Jordan for assuming the polygonal case. That view became standard. In 2007 Thomas Hales examined Jordan’s argument closely, found nothing seriously wrong with it, and noted that the polygonal case is an easy exercise. Hales also completed a computer-checked formal proof of the theorem in 2005.

In three dimensions the analogue of Schoenflies fails: James Waddell Alexander’s horned sphere of 1924 is an embedded sphere whose outside is not simply connected. The separation statement itself extends to all dimensions, as the Jordan–Brouwer theorem proved by L. E. J. Brouwer in 1911.

Professor, editor, academician, 1873–1912

In 1873 Jordan became an examiner at the École Polytechnique, and on 25 November 1876 professor of analysis there. From 1883 he also held a chair at the Collège de France. He remained, at least in name, an engineer until 1885. His lectures were known for their difficulty and for his idiosyncratic notation.

His Cours d’analyse de l’École Polytechnique appeared in three volumes between 1882 and 1887, with a much revised second edition from 1893 and a third from 1909 to 1915. It set a new standard for rigour in French analysis. The second edition, with its treatment of content, length and bounded variation, strongly influenced Lebesgue and his generation.

On 4 April 1881 he was elected to the Académie des Sciences, succeeding Michel Chasles. In 1885 he succeeded Henri Résal as editor of the Journal de Mathématiques Pures et Appliquées, the journal Liouville had founded, and edited it for 37 years until his death. He was made an officer of the Légion d’honneur in 1890, an honorary member of the London Mathematical Society in 1907, and vice-president of the International Congress of Mathematicians in Rome in 1908. He retired from teaching in 1912.

War, death and legacy, 1912–1922

Jordan’s last years were marked by loss. A daughter died in 1912. In the first two years of the First World War three of his six sons were killed, two in 1914 and a third who died of his wounds in 1915, and a grandson was killed in 1916. His wife died in 1918. Of his surviving sons, Édouard Jordan became a professor of history at the Sorbonne.

He served as president of the Académie des Sciences in 1916 and was elected a Foreign Member of the Royal Society in 1919. In September 1920 he was honorary president of the International Congress of Mathematicians in Strasbourg, the first congress after the war. His last paper, published in his own journal in 1920, returned to non-self-intersecting paths. He died in Paris on 22 January 1922, aged 84. His collected works were published in four volumes in 1961–64.

Legacy

• Group theory: the Traité made Galois theory a theory of groups. Composition series, solvable groups, linear groups over finite fields and the first infinite families of finite simple groups all begin there. The classification of finite simple groups is the completion of his programme.
• Linear algebra: the Jordan normal form and the singular value decomposition.
• Analysis: bounded variation, Jordan content and the rigorous Cours d’analyse prepared the ground for Lebesgue.
• Topology: homotopy of curves on surfaces and the Jordan curve theorem.
• Influence: Lie and Klein carried his group-theoretic view of geometry into the theory of Lie groups and the Erlangen programme.
• Honours: the Institut Camille Jordan, the mathematics research institute of the universities of Lyon, and the asteroid 25593 Camillejordan bear his name.

Several things named after a Jordan are not his. Gauss–Jordan elimination honours the geodesist Wilhelm Jordan, and Jordan algebras the physicist Pascual Jordan.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Marie Ennemond Camille Jordan, MacTutor History of Mathematics.
2. Obituary of Camille Jordan, Proceedings of the London Mathematical Society (1922), via MacTutor.
3. J. Dieudonné, Jordan, Camille, Dictionary of Scientific Biography.
4. Camille Jordan, Wikipedia.
5. H. Lebesgue, Notice sur la vie et les travaux de Camille Jordan, Mémoires de l’Académie des sciences 58 (1923).
6. F. Brechenmacher, La controverse de 1874 entre Camille Jordan et Leopold Kronecker, Revue d’histoire des mathématiques 13 (2007).

Mathematics

1. C. Jordan, Traité des substitutions et des équations algébriques, Gauthier-Villars, 1870.
2. C. Jordan, Sur la déformation des surfaces, J. Math. Pures Appl. (2) 11 (1866).
3. C. Jordan, Mémoire sur les groupes de mouvements, Annali di Matematica (2) 2 (1868–69).
4. C. Jordan, Sur la série de Fourier, C. R. Acad. Sci. Paris 92 (1881).
5. C. Jordan, Cours d’analyse de l’École Polytechnique, 3 vols., Gauthier-Villars, 1882–87; 2nd ed. 1893–96.
6. C. Jordan, Œuvres, 4 vols., Gauthier-Villars, 1961–64.
7. J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. 40 (2003).
8. T. C. Hales, Jordan’s proof of the Jordan curve theorem, Studies in Logic, Grammar and Rhetoric 10 (2007).
9. Jordan curve theorem, Bounded variation, Jordan–Schur theorem, Jordan measure, Wikipedia.
10. Jordan–Hölder theorem, Encyclopedia of Mathematics.