Ferdinand Georg Frobenius
Born: 26 Oct 1849 in Berlin, Prussia (now Germany)
Died: 3 Aug 1917 in Berlin, Germany
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Georg Frobenius is best known for his work in group theory.
He received his doctorate from the University of Berlin in 1870 after working under Weierstrass. In 1892, after having taught at the Eidgenössische Polytechnikum Zürich, he returned to Berlin to become professor of mathematics.
In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups. He published †ber Gruppen von vertauschbaren Elementen in 1879 which looks at permutable elements in groups. He collaborated with Issai Schur in representation theory of groups and character theory of groups.
His paper Über die Gruppencharactere on group characters is of fundamental importance. It was presented to the Berlin Academy on July 16 1896 and it contains work which Frobenius had done in the preceeding few months. In a series of letters to Dedekind, the first on 12 April 1896, his ideas on group characters quickly develop. Ideas from a paper by Dedekind in 1885 make an important contribution and Frobenius is able to construct a complete set of representations by complex numbers.
In a letter to Dedekind on 26 April 1896 Frobenius finds the irreducible characters for the alternating groups A
, A
, the symmetric groups S , S and the group PSL(2,7) of order 168.
Only in 1897 did Frobenius learn of Molien's work. He reformulated his work in terms of matrices and then showed that his characters are the traces of the irreducible representations. This work was published in 1897. Frobenius's character theory was used with great effect by Burnside and was beautifully written up in Burnside's 1911 edition of his Theory of Groups of Finite Order .
His representation theory for finite groups later found important applications in quantum mechanics.
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