Ferdinand Georg Frobenius

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

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.

Biography

Ferdinand Georg Frobenius was born on October 26, 1849 in Charlottenburg, a suburb of Berlin from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven. In 1867, after graduating, he went to the University of Göttingen where he began his university studies but he only studied there for one semester before returning to Berlin, where he attended lectures by Kronecker, Kummer and Karl Weierstrass. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis, supervised by Weierstrass, was on the solution of differential equations. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics. Frobenius was only in Berlin 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. He married there, brought up his family and did much important work in widely differing areas of mathematics. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. 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. 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 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 xⁿ = 1 in G is equal to kn for some positive integer k. He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equation xⁿ = 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 the classification of finite simple groups, was this problem 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. A group G is said to be a Frobenius group if there is a subgroup H < G such that

$H\cap H^x=\{1\}$ for all $x\in G-H$.

In that case, the set

$N=G-\bigcup_{x\in G-H}H^x$

together with the identity element of G forms a subgroup which is nilpotent as Thompson showed in his PhD thesis. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed the character table of the group $PSL(2,p)$ of order (1/2)(p³ − p) for all odd primes p (this group is simple provided p > 3). He also made fundamental contributions to the representation 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 determinant theorem
• 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

Publications

• Frobenius, Ferdinand Georg (1968), Serre, J.-P., ed., Gesammelte Abhandlungen. Bände I, II, III, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04120-7, MR0235974 

References

• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5, MR1715145, http://books.google.com/books?isbn=0821826778  Review

External links

• O'Connor, John J.; Robertson, Edmund F., "Ferdinand Georg Frobenius", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Frobenius.html .
• Ferdinand Georg Frobenius at the Mathematics Genealogy Project.