His 1872 Erlangen program classified geometries by their basic symmetry groups and was an influential synthesis of much of the mathematics of the time.

During his tenure at the University of Göttingen, Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes.

Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had relocated to Göttingen in 1868.

For a brief time he served as a medical orderly in the Prussian army before being appointed Privatdozent (lecturer) at Göttingen in early 1871.

This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932.

Founded by Clebsch, it grew under Klein's management, to rival, and eventually surpass Crelle's Journal, based at the University of Berlin.

Klein established a small team of editors who met regularly, making decisions in a democratic spirit.

In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the World's Columbian Exposition.

He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, by Grace Chisholm Young, an English student of Arthur Cayley's, whom Klein admired.

In 1905, he was instrumental in formulating a plan recommending that analytic geometry, the rudiments of differential and integral calculus, and the function concept be taught in secondary schools.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen program (1872), profoundly influenced the evolution of mathematics.

This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion.

They have become so much part of mathematical thinking that it is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries.

In 1879, he examined the action of PSL(2,7), considered as an image of the modular group, and obtained an explicit representation of a Riemann surface now termed the Klein quartic.

He showed that it was a complex curve in projective space, that its equation was x3y + y3z + z3x = 0, and that its group of symmetries was PSL(2,7) of order 168.

Building on methods of Charles Hermite and Leopold Kronecker, he produced similar results to those of Brioschi and later completely solved the problem by means of the icosahedral group.

Poincaré had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men.

[16] Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years.