Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin. His thesis was on the solution of differential equations, under the direction of Weierstrass. After its completion in 1870, he taught in Berlin for a few years before receiving an appointment at the Polytechnicum in Zurich (now ETH Zurich). In 1893 he returned to Berlin, and was also elected to the Prussian Academy of Sciences.
Group Theory was one of Frobenius' principal interests in the second half of his career. One of his first notable contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups) is the one still used today.
More important was his creation of the theory of group characters and group representations, which are fundamental tools for studying the structure of groups. This work led to the establishment of the Frobenius reprocity law and the definition of what are now called Frobenius groups.Contributions to Group Theory