**Georg Frobenius** combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups.

- Georg was born in Charlottenburg which was a district of Berlin which was not incorporated into the city until 1920.
- For the description of Frobenius's career so far, the attentive reader may have noticed that no mention has been made of him receiving his habilitation before being appointed to a teaching position.
- Frobenius was only in Berlin for a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum.
- For seventeen years, between 1875 and 1892, Frobenius worked in Zürich.
- We shall discuss some of the topics which he worked on below, but for the moment we shall continue to describe how Frobenius's career developed.
- Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed.
- However, for reasons which we shall discuss in a moment, Frobenius turned out to be something of a mixed blessing for mathematics at the University of Berlin.
- He describes the strained relationships which developed between Frobenius and his colleagues at Berlin.
- suspected at every opportunity a tendency of the Ministry to lower the standards at the University of Berlin, in the words of Frobenius, to the rank of a technical school ...
- Frobenius was the leading figure, on whom the fortunes of mathematics at Berlin university rested for 25 years.
- This is especially true of Frobenius.
- This was a time when there was competition between mathematians in the University of Berlin and in the University of Göttingen, but it was a competition that Göttingen won, for there mathematics flourished under Klein, much to Frobenius's annoyance.
- Frobenius hated the style of mathematics which Göttingen represented.
- Frobenius, as we said above, had extremely traditional views.
- To gain an impression of the quality of Frobenius's work before the time of his appointment to Berlin in 1892 we can do no better than to examine the recommendations of Weierstrass and Fuchs when Frobenius was elected to the Prussian Academy of Sciences in 1892.
- 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.
- The proof which Frobenius gives is the one, based on conjugacy classes, still used today in most undergraduate courses.
- In his next paper in 1887 Frobenius continued his investigation of conjugacy classes in groups which would prove important in his later work on characters.
- It was in the year 1896, however, when Frobenius was professor at Berlin that his really important work on groups began to appear.
- This paper on group characters was presented to the Berlin Academy on July 16 1896 and it contains work which Frobenius had undertaken in the preceding few months.
- Ideas from a paper by Dedekind in 1885 made an important contribution and Frobenius was able to construct a complete set of representations by complex numbers.
- It is worth noting, however, that although we think today of Frobenius's paper on group characters as a fundamental work on representations of groups, Frobenius in fact introduced group characters in this work without any reference to representations.
- In was not until the following year that representations of groups began to enter the picture, and again it was a concept due to Frobenius.
- Over the years 1897-1899 Frobenius published two papers on group representations, one on induced characters, and one on tensor product of characters.
- In 1898 he introduced the notion of induced representations and the Frobenius Reciprocity Theorem.
- He continued his applications of character theory in papers of 1900 and 1901 which studied the structure of Frobenius groups.
- Only in 1897 did Frobenius learn of Molin's work which he described in a letter to Dedekind as "very beautiful but difficult".
- 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.
- Frobenius had a number of doctoral students who made important contributions to mathematics.
- Frobenius collaborated with Schur in representation theory of groups and character theory of groups.
- It is certainly to Frobenius's credit that he so quickly spotted the genius of his student Schur.
- Frobenius's representation theory for finite groups was later to find important applications in quantum mechanics and theoretical physics which may not have entirely pleased the man who had such "pure" views about mathematics.
- Among the topics which Frobenius studied towards the end of his career were positive and non-negative matrices.
- The fact so many of Frobenius's papers read like present day text-books on the topics which he studied is a clear indication of the importance that his work, in many different areas, has had in shaping the mathematics which is studied today.
- In fact, Frobenius tried to solve mathematical problems to a large extent by means of a calculative, algebraic approach.
- For Frobenius, conceptual argumentation played a somewhat secondary role.

Born 26 October 1849, Berlin-Charlottenburg, Prussia (now Germany). Died 3 August 1917, Berlin, Germany.

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Algebra, Group Theory, Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive