A picture of Frobenius

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... is the 299th day of the year (300th in leap years) in the Gregorian calendar. ... 1849 was a common year starting on Monday (see link for calendar). ... is the 215th day of the year (216th in leap years) in the Gregorian calendar. ... 1917 (MCMXVII) was a common year starting on Monday of the Gregorian calendar (see link for calendar) or a common year starting on Tuesday of the 13-day slower Julian calendar (see: 1917 Julian calendar). ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Group theory is that branch of mathematics concerned with the study of groups. ... In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...

Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin. His thesis was on the solution of differential equations, under the direction of Weierstrass. 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. Location of Charlottenburg in Berlin Charlottenburg palace Charlottenburg is an area of Berlin within the borough of Charlottenburg-Wilmersdorf. ... This article is about the capital of Germany. ... There is no institution called the University of Berlin, but there are four universities in Berlin, Germany: Humboldt University of Berlin (Humboldt-Universität zu Berlin) Technical University of Berlin (Technische Universität Berlin) Free University of Berlin (Freie Universität Berlin) Berlin University of the Arts (Universität der... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... 1870 (MDCCCLXX) was a common year starting on Saturday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Monday of the 12-day slower Julian calendar). ... Location within Switzerland   Zürich[?] (German pronunciation IPA: ; usually spelled Zurich in English) is the largest city in Switzerland (population: 366,145 in 2004; population of urban area: 1,091,732) and capital of the canton of Zürich. ... The ETH Zurich, often called Swiss Federal Institute of Technology, is a science and technology university in the city of Zurich, Switzerland. ... Year 1893 (MDCCCXCIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day slower Julian calendar). ...

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. Group theory is that branch of mathematics concerned with the study of groups. ... The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the...

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 groups. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... // Basic definitions In mathematics, the character of a group representation of a group G is the function which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). ... 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. ... // Basic definitions In mathematics, the character of a group representation of a group G is the function which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). ...

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. In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ...

See also

In mathematics, a Frobenius algebra is an associative algebra A defined over a field K equipped with a special kind of bilinear form , then called a Frobenius form of the algebra. ... In commutative algebra and field theory, which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of rings with prime characteristic p, a class importantly including fields. ... In mathematics, the Frobenius method describes a way to find an infinite series solution for a second-order ordinary differential equation of the form We can divide through by z2 to obtain a differential equation of the form which we can solve with regular power series methods if p(z... In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ... In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix. ... In mathematics, the additive polynomials are an important topic in classical algebraic number theory. ... Wikipedia does not have an article with this exact name. ... In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ... In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite dimensional associative division algebras over the real numbers. ... In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... Burnsides lemma, sometimes also called Burnsides counting theorem, Polyas formula or Cauchy-Frobenius lemma, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ... In mathematics, the Perron–Frobenius theorem, named after Oskar Perron and Ferdinand Georg Frobenius, is a theorem in matrix theory about the eigenvalues and eigenvectors of a real positive n×n matrix: Let A = (aij) be a real n×n matrix with positive entries . ...

External links

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