Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds.

Biography

Ferdinand Georg Frobenius was born on 26 October 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. However, he studied there for only one semester before returning to Berlin, where he attended lectures by Leopold Kronecker, Ernst Kummer and Karl Weierstrass. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis 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. 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 is said to be a Frobenius group if there is a subgroup such that

:<math>\ H\ \cap\ H^x = \{\ 1\ \}\quad </math> for all <math>\quad x \in G\ \backslash\ H ~.</math>

In that case, the set

:<math>\ N = G\ \backslash \!\!\!\bigcup_{x\in\ G\ \backslash\ H}\!\!\! H^x\ </math>

together with the identity element of forms a subgroup which is nilpotent as John G. Thompson showed in 1959. 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 <math>\ \mathrm{PSL}(2,p)\ </math> of order <math>\ \tfrac{1}{2}\left(\ p^3 - p\ \right)\ </math> for all odd (this is a simple group 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)&nbsp;=&nbsp;x<sup>p</sup>&nbsp;(mod&nbsp;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&nbsp;mod&nbsp;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.

Differential equations

Frobenius made important contributions to the solution

of linear variable-coefficient ordinary differential equations

by elaborating on the resolution of power-series methods

applied at singular-points where standard Taylor-series methods fail. His algorithm is now referred to as the Frobenius method.

See also

  • List of things named after Ferdinand Georg Frobenius

Publications

  • De functionum analyticarum unius variabilis per series infinitas repraesentatione (in Latin), Dissertation, 1870
  • Über die Entwicklung analytischer Functionen in Reihen, die nach gegebenen Functionen fortschreiten (in German), Journal für die reine und angewandte Mathematik 73, 1–30 (1871)
  • Über die algebraische Auflösbarkeit der Gleichungen, deren Coefficienten rationale Functionen einer Variablen sind (in German), Journal für die reine und angewandte Mathematik 74, 254–272 (1872)
  • Über den Begriff der Irreductibilität in der Theorie der linearen Differentialgleichungen (in German), Journal für die reine und angewandte Mathematik 76, 236–270 (1873)
  • Über die Integration der linearen Differentialgleichungen durch Reihen (in German), Journal für die reine und angewandte Mathematik 76, 214–235 (1873)
  • Über die Determinante mehrerer Functionen einer Variablen (in German), Journal für die reine und angewandte Mathematik 77, 245–257 (1874)
  • Über die Vertauschung von Argument und Parameter in den Integralen der linearen Differentialgleichungen (in German), Journal für die reine und angewandte Mathematik 78, 93–96 (1874)
  • Anwendungen der Determinantentheorie auf die Geometrie des Maaßes (in German), Journal für die reine und angewandte Mathematik 79, 185–247 (1875)
  • Über algebraisch integrirbare lineare Differentialgleichungen (in German), Journal für die reine und angewandte Mathematik 80, 183–193 (1875)
  • Über das Pfaffsche Problem (in German), Journal für die reine und angewandte Mathematik 82, 230–315 (1875)
  • Über die regulären Integrale der linearen Differentialgleichungen (in German), Journal für die reine und angewandte Mathematik 80, 317–333 (1875)
  • Note sur la théorie des formes quadratiques à un nombre quelconque de variables (in French), Comptes rendus de l'Académie des sciences Paris 85, 131–133 (1877)
  • Zur Theorie der elliptischen Functionen (in German), Journal für die reine und angewandte Mathematik 83, 175–179 (1877)
  • Über adjungirte lineare Differentialausdrücke (in German), Journal für die reine und angewandte Mathematik 85, 185–213 (1878)
  • Über lineare Substitutionen und bilineare Formen (in German), Journal für die reine und angewandte Mathematik 84, 1–63 (1878)
  • Über homogene totale Differentialgleichungen (in German), Journal für die reine und angewandte Mathematik 86, 1–19 (1879)
  • Ueber Matrizen aus nicht negativen Elementen (in German), Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 26, 456—477 (1912)

References

  • Review
  • G. Frobenius, "Theory of hypercomplex quantities" (English translation)