Ferdinand Georg Frobenius
Ferdinand Georg Frobenius | |
---|---|
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 (26 October 1849 – 3 August 1917) was a
Biography
Ferdinand Georg Frobenius was born on 26 October 1849 in
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.[3] 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
- for all .
In that case, the set
together with the identity element of G forms a subgroup which is nilpotent as John G. Thompson showed in 1959.[4] 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 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
Contributions to number theory
Frobenius introduced a canonical way of turning primes into
See also
Publications
- Frobenius, Ferdinand Georg (1968), Serre, J.-P. (ed.), Gesammelte Abhandlungen. Bände I, II, III, Berlin, New York: MR 0235974
- De functionum analyticarum unius variabilis per series infinitas repraesentatione (in Latin), Dissertation, 1870
- Journal für die reine und angewandte Mathematik73, 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)
- Comptes rendus de l'Académie des sciencesParis 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
- ^ "Born in Berlin". October 26, 2010.
- ^ a b "Biography". 26 October 2010.
- S2CID 120848984.
External links
- O'Connor, John J.; Robertson, Edmund F., "Ferdinand Georg Frobenius", MacTutor History of Mathematics Archive, University of St Andrews
- G. Frobenius, "Theory of hypercomplex quantities" (English translation)