Georg Frobenius

How to use this course

Georg Frobenius’s name appears in more corners of mathematics than almost any other: the Frobenius method for differential equations, the Frobenius integrability theorem, the Frobenius automorphism, Frobenius reciprocity, Frobenius groups, the Perron–Frobenius theorem, the Frobenius norm. Behind this scattering lies a consistent style. Frobenius took problems that others had solved by calculation in special cases and found the algebraic structure that settled them in general. His greatest creation, the theory of group characters, began as an attempt to factor a single determinant.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his career: analysis in the 1870s, linear algebra and abelian groups around 1878–1880, finite groups and characters from 1887 to 1906, and nonnegative matrices from 1908 to 1912.

Timeline

Berlin, 1849–1870

Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, then a separate town west of Berlin; it was incorporated into the city only in 1920. His father, Christian Ferdinand Frobenius, was a Protestant pastor, and his mother was Christine Elizabeth Friedrich. He was known by his second name, Georg.

In 1860, aged eleven, he entered the Joachimsthal Gymnasium, one of the leading classical schools in Berlin, and graduated in 1867. He enrolled at the University of Göttingen but stayed only one semester before returning to Berlin.

The Berlin school

Berlin was then the strongest centre of mathematics in Germany. Three men dominated it. Karl Weierstrass taught analysis built on power series and rigorous estimates. Ernst Kummer taught number theory, with his ideal numbers for cyclotomic fields. Leopold Kronecker, not yet a professor but entitled to lecture as an Academy member, pursued algebra and arithmetic with an insistence on explicit, finite constructions. Frobenius absorbed all three: Weierstrass’s rigour, Kummer’s arithmetic and Kronecker’s algebra.

He received his doctorate with distinction in 1870, with a dissertation written in Latin, De functionum analyticarum unius variabilis per series infinitas repraesentatione, on representing analytic functions of one variable by infinite series. Weierstrass supervised it; the Mathematics Genealogy Project also lists Kummer as an advisor.

Schoolteacher and young professor, 1870–1875

After the doctorate Frobenius passed the examination for secondary-school teachers and taught at his old school, the Joachimsthal Gymnasium, and from 1871 at the Sophienrealschule. He kept up a steady output of research on linear differential equations, a subject then being reshaped by Lazarus Fuchs’s work on regular singular points. His 1873 paper on integrating linear differential equations by series gave the method now taught to every engineering student (Lesson 04).

In 1874 the University of Berlin appointed him extraordinary professor. This was unusual: he had not completed the habilitation that German universities normally required. Weierstrass’s strong support is the usual explanation. A year later, in 1875, the Eidgenössisches Polytechnikum in Zürich (now ETH Zürich) offered him a full professorship, and he accepted.

Zürich, 1875–1892

Frobenius spent seventeen years in Zürich. In 1876 he married Auguste Lehmann, and they raised their family there. The Zürich years were the most varied of his career. He wrote on the Pfaff problem (1877), on bilinear forms and matrices (1878–1880), on theta functions and elliptic functions, on continued fractions and the rational approximations now called Padé approximants (1881), and on finite groups.

Two collaborations and contacts stand out. With Ludwig Stickelberger, a Privatdozent at the Polytechnikum, he wrote the 1879 paper that established the structure of finite abelian groups, and later papers on elliptic functions (the Frobenius–Stickelberger formulae). With Dedekind, in Braunschweig, he began an exchange on algebra and number theory that would matter greatly in 1896.

In 1880 he wrote a paper connecting the splitting of primes in number fields with the elements of the Galois group. He did not publish it for sixteen years; it appeared in 1896. It contains the density theorem of Lesson 07 and the first systematic use of what is now called the Frobenius element. In 1887 he gave a proof of Sylow’s theorem that used only the abstract structure of a group, not its realization by permutations.

By 1892 he was one of the leading algebraists in Europe. When Kronecker died at the end of December 1891, the Berlin chair fell vacant, and Weierstrass pressed for Frobenius to fill it.

Series solutions at singular points

For the general reader. Many equations of physics, such as Bessel’s equation for a vibrating drum, have a point where the coefficients blow up. Ordinary power series fail there. Frobenius showed in 1873 how to modify the series with a fractional power and, when needed, a logarithm, and how to predict exactly which form appears.

Consider

with \(p,q\) analytic for \(|x|\lt R\). The point \(x=0\) is a regular singular point. Fuchs had characterized such points in 1866. The indicial polynomial is \(F(r)=r(r-1)+p_0r+q_0\).

Frobenius’s method

Substituting \(y=x^r\sum a_nx^n\) gives the recurrence

Frobenius’s idea was to keep \(r\) as a variable. Solve the recurrence with \(a_0=1\) for the coefficients \(a_n(r)\), rational functions of \(r\), and set \(y(x,r)=x^r\sum a_n(r)x^n\). Then all equations except the first are satisfied, and

If \(r_1\) is a double root, then \(F(r_1)=F'(r_1)=0\), so differentiating in \(r\) gives \(L\big[\partial_ry(x,r)\big|_{r=r_1}\big]=0\). Since \(\partial_rx^r=x^r\ln x\), the second solution contains \(y_1\ln x\). For roots differing by an integer, one multiplies first by \(r-r_2\) to clear the pole of \(a_n(r)\) and then differentiates. Convergence follows by comparing with a geometric series, using \(|F(r+n)|\sim n^2\).

The Pfaff problem and integrability

For the general reader. At each point of space, pick a plane. Is there a family of surfaces tangent to all the planes? Usually not: the planes may twist like the pages of a spiralling book. Frobenius gave the exact test, which today is a basic theorem of differential geometry and of control theory.

A Pfaffian equation is \(\omega=0\) with \(\omega=\sum_ia_i(x)\,dx_i\) a differential 1-form. The Pfaff problem, posed by Johann Friedrich Pfaff in 1815, asks for the smallest number of relations among the variables that satisfy it. Frobenius’s paper of 1877 gave a complete algebraic treatment. Earlier partial versions of the integrability criterion are due to Feodor Deahna (1840) and Alfred Clebsch (1866).

Why the condition is necessary

If \(\omega^i=\sum_jg^i_j\,df_j\) with \((g^i_j)\) invertible, then \(d\omega^i=\sum_jdg^i_j\wedge df_j\), and each \(df_j\) is a combination of the \(\omega^k\). This gives condition 2. Sufficiency is proved by induction on dimension, or in the vector-field form by straightening commuting flows. For \(r=1\) in \(\mathbb R^3\), with \(\omega=P\,dx+Q\,dy+R\,dz\) and \(\mathbf F=(P,Q,R)\), the condition reads

Normal forms

For a single form, Frobenius defined its class, the least number of variables in which it can be written after a change of coordinates, and showed that a form of constant class has a local normal form: \(\sum_{i=1}^m x_i\,dy_i\) for class \(2m\), and \(dz+\sum_{i=1}^m x_i\,dy_i\) for class \(2m+1\). Gaston Darboux reached equivalent results in 1882, and the statement is often called Darboux’s theorem. The odd normal form \(dz-y\,dx\) in \(\mathbb R^3\) is the standard contact form.

Bilinear forms, matrices and division algebras

For the general reader. A matrix is a table of numbers that can be multiplied like a number. In 1878 Frobenius wrote the first general theory of this multiplication: he proved that every square matrix satisfies its own characteristic equation and found the smallest equation it satisfies. In the same paper he showed that the real numbers, the complex numbers and the quaternions are the only number systems of their kind.

Frobenius wrote in the language of bilinear forms \(\sum a_{ij}x_iy_j\), composed by substitution, which is matrix multiplication. Arthur Cayley had stated in 1858 that a matrix satisfies its characteristic equation, checking it for \(2\times2\) matrices and asserting it for \(3\times3\). William Rowan Hamilton had a version for quaternions.

Proof. Let \(B(t)=\operatorname{adj}(tI-A)\), a matrix whose entries are polynomials of degree \(\le n-1\). Write \(B(t)=\sum_{k=0}^{n-1}B_kt^k\). From \((tI-A)B(t)=\chi_A(t)I=\sum c_kt^kI\), compare coefficients: \(B_{k-1}-AB_k=c_kI\). Multiply the \(k\)-th equation by \(A^k\) on the left and sum: the left side telescopes to 0, and the right side is \(\chi_A(A)\). The minimal polynomial exists because \(\{f:f(A)=0\}\) is a non-zero ideal of \(K[t]\), and every ideal of \(K[t]\) is principal.

Proof sketch

Let \(D\) be such an algebra, of dimension \(n\). For \(x\in D\), the minimal polynomial of \(x\) over \(\mathbb R\) is irreducible, because \(D\) has no zero divisors, so it has degree 1 or 2. Hence \(x=a+v\) with \(a\in\mathbb R\) and \(v^2\in\mathbb R_{\le0}\). The set \(V=\{v: v^2\in\mathbb R_{\le0}\}\) is a subspace of dimension \(n-1\), and \(\langle u,v\rangle=-\tfrac12(uv+vu)\) is a positive definite inner product on it. If \(n\gt1\), choose a unit \(i\in V\); if \(n\gt2\), choose a unit \(j\in V\) orthogonal to \(i\), so that \(ij=-ji\) and \(k=ij\in V\). Then \(\mathbb R+\mathbb Ri+\mathbb Rj+\mathbb Rk\cong\mathbb H\), Now let \(w\in V\) be orthogonal to \(i,j,k\). Orthogonal elements of \(V\) anticommute, so \(wi=-iw\) and \(wj=-jw\), hence \(wk=w\,ij=ij\,w=kw\). But \(w\perp k\) also gives \(wk=-kw\). So \(wk=0\), and since \(D\) has no zero divisors, \(w=0\). Hence \(n\in\{1,2,4\}\). Charles Sanders Peirce published a similar result independently in 1881.

Frobenius’s work of 1878–1880 also covered the equivalence and similarity of matrices: the theory of elementary divisors (building on Weierstrass and Kronecker), invariant factors of integer matrices, and the rational canonical form, which is often named after him.

Abstract groups and the density theorem

For the general reader. In the 1870s groups were still mostly groups of permutations. Frobenius was one of the first to treat a group purely as a set with a multiplication, and to prove theorems from that alone. He also saw that the Galois group of a polynomial controls how the polynomial factors modulo each prime, and predicted how often each pattern occurs.

Kronecker had proved a decomposition in 1870. Frobenius and Stickelberger proved uniqueness and treated the subject abstractly, with application to Kummer’s ideal classes and to residues modulo \(m\).

Proof sketch. Induct on \(|G|\). If \(p\) divides \(|Z(G)|\), the centre has an element \(z\) of order \(p\) (Cauchy’s theorem for abelian groups); a subgroup of order \(p^{a-1}\) in \(G/\langle z\rangle\) pulls back to one of order \(p^a\). Otherwise the class equation \(|G|=|Z(G)|+\sum[G:C_G(x_i)]\) shows some non-central class has index prime to \(p\). Then \(p^a\) divides \(|C_G(x_i)|\lt|G|\), and induction applies.

In 1895 Frobenius proved a far-reaching generalization: if \(n\) divides \(|G|\), the number of solutions of \(x^n=1\) in \(G\) is a multiple of \(n\). He conjectured that if there are exactly \(n\) solutions they form a subgroup. That conjecture was settled only in 1991, using the classification of finite simple groups.

The density theorem

Let \(f\in\mathbb Z[x]\) be monic, irreducible of degree \(n\), with Galois group \(G\le S_n\) acting on its roots. For a prime \(p\) not dividing the discriminant, \(f\bmod p\) factors into distinct irreducibles whose degrees form a partition of \(n\), the splitting type of \(p\). Frobenius attached to such \(p\) a conjugacy class of \(G\), the Frobenius elements \(\sigma_p\), whose cycle type equals the splitting type.

Frobenius conjectured the sharper statement that each conjugacy class \(C\) occurs as the class of \(\sigma_p\) for primes of density \(|C|/|G|\). Nikolai Chebotarev proved it in 1922. For cyclotomic polynomials it reduces to Peter Gustav Lejeune Dirichlet’s theorem on primes in arithmetic progressions.

The group determinant and characters

For the general reader. In 1896 Dedekind told Frobenius about a puzzle. Write a variable for each element of a finite group and form a certain determinant. For commutative groups it splits into simple linear factors; for others it does not, and Dedekind could not see why. Frobenius solved the puzzle in a few months, and the solution was a new theory: the characters of a group, the tool that later became indispensable in quantum mechanics and in the classification of finite simple groups.

Let \(G\) be a finite group and \(\{x_g:g\in G\}\) independent variables. The group determinant is

For \(G\) abelian, Dedekind had shown that \(\Theta\) is a product of \(|G|\) linear factors \(\sum_g\chi(g)x_g\), one for each homomorphism \(\chi:G\to\mathbb C^\times\). He had computed non-abelian examples such as \(S_3\) and found irreducible factors of higher degree. In an exchange of letters beginning in April 1896, he described the problem to Frobenius.

Frobenius defined the characters \(\chi_i\) through the coefficients of the \(\Phi_i\) and proved their orthogonality in his paper presented to the Berlin Academy on 16 July 1896. In 1897 he showed that \(\chi_i(g)=\operatorname{tr}\rho_i(g)\) for an irreducible matrix representation \(\rho_i:G\to\mathrm{GL}_{f_i}(\mathbb C)\), and that \(\Phi_i=\det\big(\sum_gx_g\rho_i(g)\big)\). This is the birth of representation theory. Theodor Molien, working on algebras, reached part of the theory independently, and William Burnside soon added his own approach.

Proof sketch (in the form later given by Schur). For any matrix \(M\), the average \(\tilde M=\frac1{|G|}\sum_g\rho_i(g)M\rho_j(g^{-1})\) intertwines \(\rho_j\) with \(\rho_i\). Schur’s lemma says \(\tilde M=0\) if \(i\ne j\), and \(\tilde M=\frac{\operatorname{tr}M}{f_i}I\) if \(i=j\). Taking matrix units for \(M\) and then traces gives the first relation. The second follows because the character table is a square matrix, and the first relation says it becomes unitary after weighting columns by \(\sqrt{|C|/|G|}\).

Induction, symmetric groups, Frobenius groups

For the general reader. Having built the theory, Frobenius used it. He showed how representations of a small group generate those of a larger one, computed every character of the symmetric groups with one formula, and proved a theorem about the structure of certain groups for which, more than a century later, no proof without characters is known.

For a subgroup \(H\le G\) and a class function \(\psi\) on \(H\), the induced class function is

Characters of the symmetric group (1900)

Irreducible characters \(\chi^\lambda\) of \(S_n\) are indexed by partitions \(\lambda=(\lambda_1\ge\cdots\ge\lambda_k)\) of \(n\). For a permutation of cycle type \(\mu\), let \(p_\mu=\prod_jp_{\mu_j}\) with power sums \(p_m=x_1^m+\cdots+x_k^m\), and let \(\Delta=\prod_{i\lt j}(x_i-x_j)\).

Frobenius groups (1901)

The proof extends characters of \(H\) to \(G\) by induction and correction, and shows \(N\) is the intersection of their kernels. In permutation language, \(G\) acts transitively on a set with no non-identity element fixing two points, and \(N\) consists of the fixed-point-free elements with 1. John Thompson proved in 1959 that \(N\) is nilpotent.

With Schur in 1906, Frobenius introduced the indicator \(\nu(\chi)=\frac1{|G|}\sum_g\chi(g^2)\), which equals \(1\), \(0\) or \(-1\) according as the irreducible representation is realizable over \(\mathbb R\), has a non-real character, or has a real character but is not realizable over \(\mathbb R\).

Nonnegative matrices

For the general reader. Matrices with positive entries describe populations, Markov chains, economies and web links. In 1907 Oskar Perron proved that such a matrix has a dominant positive eigenvalue with a positive eigenvector. Frobenius, near the end of his career, extended this to matrices with zeros, found the right condition (irreducibility) and described exactly what can go wrong.

A square matrix \(A\ge0\) (entrywise) is irreducible if there is no permutation matrix \(P\) with \(PAP^{\mathsf T}=\begin{pmatrix}B&C\\0&D\end{pmatrix}\), \(B,D\) square and non-empty; equivalently, the directed graph with an edge \(i\to j\) when \(a_{ij}\gt0\) is strongly connected. Let \(\rho(A)\) be the spectral radius.

Proof idea

Consider the Collatz–Wielandt function \(c(x)=\min_{i:\,x_i\gt0}(Ax)_i/x_i\) on nonnegative non-zero vectors. Irreducibility gives \((I+A)^{n-1}\gt0\), which lets one restrict to a compact set of positive vectors, where \(c\) attains a maximum \(r\). A maximizer is shown to be an eigenvector with eigenvalue \(r\), and \(|\lambda|\le r\) for every eigenvalue by applying \(c\) to \(|x|\). Simplicity and the cyclic structure come from analysing when the triangle inequality \(|Ax|\le A|x|\) is an equality. The result:

Kronecker’s chair, 1892–1917

In 1892 Frobenius returned to Berlin as Kronecker’s successor; Adolf Hurwitz took over his chair in Zürich. In the same year Hermann Amandus Schwarz succeeded Weierstrass, and with Lazarus Fuchs the three formed the Berlin professoriate for the next decades. Weierstrass and Fuchs wrote the proposal for Frobenius’s election to the Prussian Academy of Sciences, listing fifteen areas of his work, and he became a member in 1893.

Frobenius saw himself as the guardian of the Berlin tradition. He held that universities should teach pure mathematics and leave applications to the technical colleges. He was suspicious of the style of Göttingen, where Felix Klein combined mathematics with applications, organization and school reform, and he was openly hostile to the work of Klein and Sophus Lie. Colleagues described him as quick-tempered and quarrelsome. His standards were exacting, and historians have linked his influence to a decline in the number of Berlin doctorates and habilitations even as student numbers grew, while Göttingen rose to first place.

Students and the Berlin algebra school

His most important student was Issai Schur, who took his doctorate in 1901 with a thesis on polynomial representations of the general linear group and became his collaborator and successor in representation theory. Edmund Landau (1899) became a leading analytic number theorist. Other doctoral students included Richard Fuchs, Konrad Knopp (1907, jointly with Friedrich Schottky), Ernst Jacobsthal, Walter Schnee and Robert Remak. Carl Ludwig Siegel, a student in Berlin from 1915, attended his last lectures and remembered him warmly.

Death and legacy

Frobenius continued to publish to the end, with papers on nonnegative matrices (1908–1912) and on Markov numbers (1913). He died in Charlottenburg on 3 August 1917, aged 67, in the middle of the First World War. Constantin Carathéodory succeeded him in Berlin in 1918. His Gesammelte Abhandlungen, in three volumes edited by Jean-Pierre Serre, were published by Springer in 1968.

Legacy

• Representation theory: characters, induction and reciprocity, the characters of \(S_n\), and Frobenius groups. Through Schur, Emmy Noether, Richard Brauer and Hermann Weyl, the theory reached quantum mechanics and, through Brauer, the classification of finite simple groups.
• Algebraic number theory: the Frobenius element and the density theorem, which lead to Chebotarev’s theorem and, through Emil Artin, to Artin L-functions and reciprocity.
• Linear algebra: the minimal polynomial, the first general proof of Cayley–Hamilton, the rational canonical form, and the Frobenius norm \(\|A\|_F=\big(\sum|a_{ij}|^2\big)^{1/2}\).
• Analysis and geometry: the Frobenius method for differential equations, the integrability theorem, and later concepts bearing his name such as Frobenius manifolds.
• Applied mathematics: Perron–Frobenius theory, used in Markov chains, population models, economics and ranking algorithms.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Ferdinand Georg Frobenius, MacTutor History of Mathematics.
2. Ferdinand Georg Frobenius, Wikipedia (with references).
3. Ferdinand Georg Frobenius, Mathematics Genealogy Project.
4. H. Wussing, Frobenius, Georg Ferdinand, Dictionary of Scientific Biography.
5. T. Hawkins, The Mathematics of Frobenius in Context, Springer, 2013.
6. K.-R. Biermann, Die Mathematik und ihre Dozenten an der Berliner Universität 1810–1933, Akademie-Verlag, 1988.

Mathematics

1. G. Frobenius, Gesammelte Abhandlungen, ed. J.-P. Serre, 3 vols., Springer, 1968.
2. G. Frobenius, Über die Integration der linearen Differentialgleichungen durch Reihen, J. reine angew. Math. 76 (1873).
3. G. Frobenius, Über das Pfaffsche Problem, J. reine angew. Math. 82 (1877).
4. G. Frobenius, Über lineare Substitutionen und bilineare Formen, J. reine angew. Math. 84 (1878).
5. G. Frobenius and L. Stickelberger, Über Gruppen von vertauschbaren Elementen, J. reine angew. Math. 86 (1879).
6. G. Frobenius, Über Gruppencharaktere, Sitzungsber. Preuss. Akad. Wiss. (1896).
7. T. Y. Lam, Representations of finite groups: a hundred years, Notices AMS 45 (1998).
8. P. Stevenhagen and H. W. Lenstra, Chebotarëv and his density theorem, Math. Intelligencer 18 (1996).
9. C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000, ch. 8 (Perron–Frobenius theory).