**Camille Jordan** was highly regarded by his contemporaries for his work in algebra, group theory and Galois theory.

- Jordan studied at the Lycée de Lyon and at the Collège d'Oullins.
- This establishment provided training to be an engineer and Jordan, like many other French mathematicians of his time, qualified as an engineer and took up that profession.
- Cauchy in particular had been one to take this route and, like Cauchy, Jordan was able to work as an engineer and still devote considerable time to mathematical research.
- Jordan's doctoral thesis was in two parts with the first part Sur le nombre des valeurs des fonctions Ⓣ(On the number of function values) being on algebra.
- Jordan was examined on 14 January 1861 by Duhamel, Serret and Puiseux.
- The topic of the second part of Jordan's thesis had been proposed by Puiseux and it was this second part which the examiners preferred.
- Jordan was a mathematician who worked in a wide variety of different areas essentially contributing to every mathematical topic which was studied at that time.
- Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
- Jordan introduced the notion of homotopy of paths looking at the deformation of paths one into the other.
- Jordan was particularly interested in the theory of finite groups.
- This is not really an accurate statement, for it would be reasonable to argue that before Jordan began his research in this area there was no theory of finite groups.
- It was Jordan who was the first to develop a systematic approach to the topic.
- Serret, Bertrand and Hermite had attended Liouville's lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.
- To Jordan a group was what we would call today a permutation group; the concept of an abstract group would only be studied later.
- Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property).
- Jordan proved the Jordan-Hölder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.
- Jordan, however, clearly saw this as an aim of the subject, even if it was not one which might ever be solved.
- The treatise contains the 'Jordan normal form' theorem for matrices, not over the complex numbers but over a finite field.
- It would also be fair to say that group theory was one of the major areas of mathematical research for 100 years following Jordan's fundamental publication.
- Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure.
- Jordan's interest in groups of Euclidean transformations in three dimensional space influenced Lie and Klein in their own theories of continuous and discontinuous groups.
- The publication of Traité des substitutions et des équations algebraique Ⓣ(Treatise on substitutions and algebraic equations) did not mark the end of Jordan's contribution to group theory.
- Although there are infinite families of such finite subgroups, Jordan found that they were of a very specific group theoretic structure which he was able to describe.
- Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem.
- The second edition appeared in 1893 while the Jordan curve theorem appeared in the third edition of the text which appeared between 1909 and 1915.
- Of course by 1882, when the first volume was published, Jordan was lecturing at the École Polytechnique and the book was written as a text for the students there.
- However between the editions Jordan had taught more advanced courses on analysis at the Collège de France and this may have influenced him to put set topology right up front in the second edition.
- Among Jordan's many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series.
- Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.
- In 1912 Jordan retired from his positions.
- Among the honours given to Jordan was his election to the Académie des Sciences on 4 April 1881.
- The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying.
- Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980).

Born 5 January 1838, La Croix-Rousse, Lyon, France. Died 22 January 1922, Paris, France.

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Algebra, Group Theory, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive