Ferdinand Georg Frobenius

Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Born	(1849-10-26)October 26, 1849 Charlottenburg
Died	August 31, 1917(1917-08-31) (aged 67) Berlin
Nationality	German
Alma mater	University of Göttingen University of Berlin
Known for	Differential equations Group theory Cayley-Hamilton theorem
Scientific career
Fields	Mathematics
Institutions	University of Berlin ETH Zurich
Doctoral advisor	Karl Weierstrass Ernst Kummer
Doctoral students	Richard Fuchs Edmund Landau Issai Schur Konrad Knopp Walter Schnee

Ferdinand Georg Frobenius (October 26, 1849 – August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. He also gave the first full proof for the Cayley-Hamilton theorem.

Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin. His thesis, supervised by Weierstrass, was on the solution of differential equations. 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, where he was elected to the Prussian Academy of Sciences.

Contributions to group theory

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 one of those frequently used today.

• Frobenius also has proved the following fundamental theorem: If a positive integer $n$ divides the order $|G|$ of a finite group $G$, then the number of solutions of the equation $x^n=1$ in $G$ is equal to $kn$ for some positive integer $n$. He also posed the following problem: If, in the above theorem, $k=1$, then the solutions of the equation $x^n=1$ in $G$ form a subgroup. Many years ago this problem was solved for solvable groups (see textbook by Marshall Hall Jr., Theorem 9.4.1). Only in 1991, after classification of finite simple groups, this problem was solved in general.

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 notion of Frobenius reciprocity and the definition of what are now called Frobenius groups. He also made fundamental contributions to the character theory of the symmetric and alternating groups.

Contributions to number theory

Frobenius introduced a canonical way of turning primes into conjugacy classes in Galois groups over Q. Specifically, if K/Q is a finite Galois extension then to each (positive) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal(K/Q) satisfying the condition g(x) = x^p (mod P) for all integers x of K. Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p. If we take for K the mth cyclotomic field, whose Galois group over Q is the units modulo m (and thus is abelian, so conjugacy classes become elements), then for p not dividing m the Frobenius class in the Galois group is p mod m. From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of Q depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study.

See also

• Arithmetic and geometric Frobenius
• Frobenius algebra
• Frobenius endomorphism (also known as Frobenius morphism)
• Frobenius method
• Frobenius norm
• Frobenius normal form
• Frobenius number
• Frobenius polynomial
• Frobenius pseudoprime
• Frobenius theorem (differential topology)
• Frobenius theorem (real division algebras)
• Frobenius-Schur indicator
• Cauchy-Frobenius lemma
• Perron–Frobenius theorem

External links

• O'Connor, John J.; Robertson, Edmund F., "Ferdinand Georg Frobenius", MacTutor History of Mathematics Archive, University of St Andrews
• Ferdinand Georg Frobenius at the Mathematics Genealogy Project