Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
, best known for his contributions to the theory of
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
,
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, and to
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. 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 approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...
s), 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 manifold
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin,B. Dubrovin: ''Geometry of 2D topological field theories.'' In: Springer LNM, 1620 (1996), pp. 120–348. is a flat Riemannian manifold with a cer ...
s.
Biography
Ferdinand Georg Frobenius was born on 26 October 1849 in
Charlottenburg
Charlottenburg () is a Boroughs and localities of Berlin, locality of Berlin within the borough of Charlottenburg-Wilmersdorf. Established as a German town law, town in 1705 and named after Sophia Charlotte of Hanover, Queen consort of Kingdom ...
, a suburb of
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
from parents Christian Ferdinand Frobenius, a
Protestant
Protestantism is a Christian denomination, branch of Christianity that follows the theological tenets of the Reformation, Protestant Reformation, a movement that began seeking to reform the Catholic Church from within in the 16th century agai ...
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
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
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
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
. He received his doctorate (awarded with distinction) in 1870 supervised by
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
. 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.
Frobenius was only in Berlin a year before he went to
Zürich
Zürich () is the list of cities in Switzerland, largest city in Switzerland and the capital of the canton of Zürich. It is located in north-central Switzerland, at the northwestern tip of Lake Zürich. As of January 2020, the municipality has 43 ...
to take up an appointment as an ordinary professor at the
Eidgenössische Polytechnikum. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, 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
The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
.
Contributions to group theory
Group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
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
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...
for abstract groups. Earlier proofs had been for
permutation group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
s. 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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
''G'', then the number of solutions of the equation ''x''
''n'' = 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 ''x''
''n'' = 1 in ''G'' form a subgroup. Many years ago this problem was solved for
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
s. Only in 1991, after the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
, was this problem solved in general.
More important was his creation of the theory of
group characters and
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s, which are fundamental tools for studying the structure of groups. This work led to the notion of
Frobenius reciprocity
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find ...
and the definition of what are now called
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure
Suppos ...
s. 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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
as
John G. Thompson
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008.
...
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
of order (1/2)(''p''
3 − 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 mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
in
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
s 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 ''m''th
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
, 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
*
List of things named after Ferdinand Georg Frobenius
Publications
*
*''
De functionum analyticarum unius variabilis per series infinitas repraesentatione'' (in Latin), Dissertation, 1870
*''
Ãœber die Entwicklung analytischer Functionen in Reihen, die nach gegebenen Functionen fortschreiten'' (in German),
Journal für die reine und angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
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
External links
*
G. Frobenius, "Theory of hypercomplex quantities"(English translation)
{{DEFAULTSORT:Frobenius, Ferdinand Georg
1849 births
1917 deaths
19th-century German mathematicians
20th-century German mathematicians
Group theorists
Linear algebraists
Members of the Prussian Academy of Sciences
Scientists from Berlin
People from the Province of Brandenburg
University of Göttingen alumni
Humboldt University of Berlin alumni
Humboldt University of Berlin faculty
ETH Zurich faculty
People from Charlottenburg