Otto Ludwig Holder influenced math in a number of areas. His main contributions include work on the convergence of Hourier series and the uniqueness of the factor groups in a composition series.

Holder was born on December 22 in the year 1859. The aspiring mathematician, native to Stuttgart, Germany, made his mark in the math world quickly. He studied engineering at a polytechnic school in Stuttgart for one year at the young age of eighteen. From there, in 1877, he began to study at the University of Berlin. While at Berlin he was influenced by the high caliber mathematicians that he was taught by. He was a student of Runge and attended lectures by Weierstrass, Kronecker and Kummer. Holder earned his doctorate then traveled to Leipzig, Germany.

In 1882 he presented his dissertation to the University of Tubingen, which investigated the analytic functions and summation procedures by arithmetic means. Just two years later he became a lecturer at Gottingen University. While there he worked at Gottingen where he achieved what is perhaps the most influential discovery of his career. Also while at Gottingen he began work in group theory with von Dyck and Klein. Holder's career suffered a small setback in 1889. After being offered a position in Tubingen he suffered a mental collapse. The mathematician, however, made a steady recovery and, as the faculty at Tubingen kept their confidence in him, he fully recovered soon and gave his inaugural lecture in 1890.

Holder was influenced early on by Kronecker, a lecturer at the University of Berlin. Kronecker's liking for rigor had a profound influence on Holder's later work in algebra. His core of studies revolved around the convergence of Fourier series, group series, and the uniqueness of the factor groups in a composition series; he covered a much vaster area of mathematics.

Holder expressed an interest in function theory while he was at Leipzig. When he began to lecture at Gottingen he began to work on the convergence of the Fourier series and discovered the inequality that is now named after him. Holder proved the uniqueness of the factor groups in a composition series. The theorem he developed is now called the Jordan-Holder theorem.

Although Holder did not credit himself with the invention of a factor group, the concept appeared clearly for the first time in a paper published by him in 1889. His papers presented the concept with clarity although he claimed that the concept was neither new nor difficult, but was not sufficiently appreciated.

In 1891 Holder began to study the irreducible case of the cubic in the Cardan-Tartaglis formula with the aid of group theory and Galois theory methods. His contributions to group theory continued through his career. He searched for finite simple groups and in an 1892 paper he proved 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 using modern methods. He also studied groups of orders p3, pq2, pqr, and p4 for p, q, r primes. He published his results in 1894, which also heavily rely on the use of the use of Sylow theorems.

Holder wrote a paper and published in 1895, introducing concepts including inner and outer automorphisms. His paper also included the extensions of groups.

After 1900 Holder began to be interested in the geometry of the projective line and later devoted his life to studying philosophical questions.

A fellow mathematician, van der Waerden, said of Holder, "reading Holder's papers again and again is a profound intellectual treat."

While still living in Leipzig, Germany, Holder died at the age of seventy-seven on August 29, 1937. In his life Holder gained widespread acclaim and respect among mathematicians. His theorems and way of thinking developed in the 1800s were modern enough to still be in use today.