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.
Hölder studied engineering at the polytechnic in Stuttgart for a year then, from 1877, he studied at the University of Berlin. At Berlin he was a fellow student of Runge and he attended lectures by Weierstrass, Kronecker and Kummer. 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.
Hölder presented his dissertation to the University of Tübingen in 1882. His dissertation investigates analytic functions and summation procedures by arithmetic means.
After taking his doctorate Hölder went to Leipzig. Klein was there at the time but there seems to have been little interaction between the two at the time, Hölder at this time still being interested in function theory, although Klein had a strong influence on Hölder later.
Hölder became a lecturer at Göttingen in 1884 and at first he worked on the convergence of Fourier series. Shortly after be began working at Göttingen he discovered the inequality now named after him. It appears that Hölder became interested in group theory while at Göttingen, through von Dyck and Klein.
Hölder was offered a post in Tübingen in 1889 but unfortunately he suffered a mental collapse. The faculty at Tübingen kept their confidence in Hölder and he made a steady recovery, giving his inaugural lecture in 1890.
He began to study the Galois theory of equations and from there he was led to study compostion series of groups.
Although Hölder did not consider that he invented the notion of a factor group, the concept appears clearly for the first time in a paper of Hölder's of 1889. Hölder clarifies the concept which he claims is neither new nor difficult but is not sufficiently appreciated.
Hölder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Hölder theorem.
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 made many other contributions to group theory. He searched for finite simple groups and in an 1892 paper he showed that all simple groups up to order 200 are already known. His methods use the Sylow theorems in a similar way to how the problem would be solved today. Hölder also studied groups of orders p3, pq2, pqr and p4 for p, q, r primes, publishing his results in 1893, these results again heavily rely on the use of Sylow theorems.
Concepts which were introduced by Hölder include inner and outer automorphisms. In 1895 he wrote a long paper on extensions of groups.
From 1900 he became interested in the geometry of the projective line and later he studied philosophical questions.
In  van der Waerden writes:-
reading Hölder's papers again and again is a profound intellectual treat.
Article by: J J O'Connor and E F Robertson