**Otto Hölder** worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him. He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series.

- Eduard Otto Hölder studied law in Tübingen and became a professor of law at several different universities.
- Otto Hölder studied at a Gymnasium in Stuttgart, in fact at one of the earliest Gymnasiums specialising in science, graduating in 1876.
- Weierstrass made a marked impression of the young Hölder and his influence showed on Hölder throughout his career.
- Hölder's interest in algebra came partly through the influence of Kronecker at this time and Kronecker's liking for rigour almost certainly was to have a profound influence on Hölder's later work in algebra.
- After studying in Berlin, Hölder went to the Eberhard-Karls University of Tübingen where he was advised by Paul du Bois-Reymond.
- These summation procedures are now the known as the "Hölder summation method".
- His dissertation also contains the continuity condition for volume density which now is known as the "Hölder condition" on a function.
- After the award of his doctorate, Hölder went to Leipzig.
- Hölder at this time was still interested in function theory, although Klein had a strong influence on Hölder later in his career.
- Strangely, Göttingen did not recognise Hölder's Tübingen doctorate so, in 1884, he submitted a thesis for a second doctorate at Göttingen and, in the same year, habilitated at the University of Göttingen.
- It appears that Hölder became interested in group theory while at Göttingen, partly through discussions with Walther von Dyck and partly through Felix Klein who was lecturing on Galois theory.
- The university faculty at Göttingen wanted to offer Hölder an assistant lectureship but such appointments could only be made by the Prussian Ministry of Culture and, despite repeated requests, they felt that he did not have sufficient lecturing experience for such a post.
- Hölder was offered a post in Tübingen in May 1889 but unfortunately he suffered a mental collapse.
- The faculty at Tübingen were unsure how to proceed when they learnt that Hölder was ill and in a clinic but, after much discussion, they kept their confidence in Hölder.
- Klein's lectures on Galois theory at Göttingen had interested Hölder who began to study the Galois theory of equations and from there he was led to study composition series of groups.
- Hölder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Hölder theorem, and published the result in Mathematische Annalen in 1889 in the paper Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen Ⓣ(Recycle any algebraic equation in a chain of equations).
- Although Hölder did not consider that he invented the notion of a factor group, the concept appears clearly for the first time this paper of Hölder's.
- With the help of group theory and Galois theory methods Hölder returned to a study of the irreducible case of the cubic in the Cardan-Tartaglia formula in 1891.
- Hölder's proof of this result, which had long been suspect, was accompanied by three accounts by other people which appeared around the same time and were summarised in the second volume of Netto's book 'Vorlesungen über Algebra' Ⓣ(Lectures on algebra) (1900).
- Hölder made many other contributions to group theory.
- Concepts which were introduced by Hölder include inner and outer automorphisms.
- The difference between their work is that Schreier drew out each idea to its logical conclusion, whereas Hölder had been motivated initially by the wish to classify particular sorts of groups and therefore developed the theory with this fixed aim in mind.
- reading Hölder's papers again and again is a profound intellectual treat.
- In 1889 Hölder was appointed as an Extraordinary Professor of Mathematics at the University of Tübingen.
- Ernst Hölder became a mathematician working mainly in the field of mathematical physics.
- David Hilbert was ranked first, Friedrich Schottky was ranked second and Hölder was ranked third.
- From 1900 Hölder became interested in the geometry of the projective line and philosophical questions, which had interested him throughout his career, began to play a prominent role.
- Perhaps we should begin by looking at the paper by Hölder, published in 1892, in which he gives his reaction to Robert Grassmann's Die Zahlenlehre oder Arithmetik - streng wissenschaftlich in strenger Formelentwicklung Ⓣ(The theory of numbers or arithmetic - strictly scientific in strict formula development) (1891).
- Second, Hölder's analysis of Robert Grassmann's foundational ideas provides an important assessment of the contribution of Hermann and Robert Grassmann to the axiomatisation of arithmetic, a contribution which, though often mentioned, is itself still not widely acknowledged and not fully understood.
- Third, the effort of exposing the weak spots in Robert Grassmann's ideas led Hölder to formulate the main problems confronting formal axiomatics: independence of the axioms, consistency, completeness, and the issue of the relationship between pure mathematics and its applications.
- When Hölder was appointed to the chair in Leipzig in 1899, he delivered the inaugural lecture 'Anschauung und Denken in der Geometrie' which was published in 1900.
- This inaugural lecture was the starting point for Hölder's axiomatic theory of quantity which he published as Die Axiome der Quantität und die Lehre vom Mass Ⓣ(The axioms of quantity and the doctrine of mass) in 1901.
- Finally, and above all, there is required an unusual ability in order to realize the goal which Dr Hölder has set before himself, namely to bring to self-conscious expression the logic of mathematical inference.
- Dr Hölder (who is professor of mathematics at Leipzig) desires his work to be regarded as primarily a contribution to the logic of mathematics, and modestly leaves to other logicians the problem of incorporating his findings in more comprehensive treatises.

Born 22 December 1859, Stuttgart, Germany. Died 29 August 1937, Leipzig, Germany.

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Algebra, Group Theory, Origin Germany

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