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The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics

Thomas Hawkins
Publication Date: 
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Sources and Studies in the History of Mathematics and Physical Sciences
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
David P. Roberts
, on

Ferdinand Georg Frobenius was born in 1849 and spent his early career at the University of Berlin. From 1874 to 1892 he worked in Zurich at the institution now known as ETH. In 1892 he returned to the University of Berlin, working there until his death in 1917. He made fundamental contributions to many areas of mathematics, being most famous for founding the theory of representations of finite groups.

In 1968, Frobenius’ collected works came out in a three-volume set, edited by Jean-Pierre Serre. Serre writes in his one-page preface,

On ne trouvera aucune analyse des travaux de Frobenius, ni de leur influence sur les recherches ultérieures. Une telle analyse, en effet, eut été fort difficile à faire…
There is no analysis of the work of Frobenius, nor of its influence on later research. Such an analysis, in fact, would have been very difficult to do…

Hawkins has spent more than forty years on this difficult task. The present work weaves his previously published papers and much more into a description of Frobenius’ entire career. As the title suggests, approximately equal attention is spent on providing context. Thus besides describing Frobenius’ work, Hawkins regularly relates it to work of other mathematicians, earlier, contemporary, and later.

The text is divided into three parts. Chapters 1 and 2 summarize Frobenius’ life and mathematics. Chapters 3–5 set the stage by discussing some pre-Frobenius Berlin mathematics. Chapters 6–18, clearly the core of the book at five hundred pages, focus on Frobenius’ mathematical achievements in roughly chronological order.

I highly recommend Hawkins’ book. It is very mathematical all the way through. But, in terms of structure, it reads something like a large novel. There is a central protagonist but also many other sharply drawn characters of great interest. There are many appealing individual stories, and they all interlock into a satisfying whole.

To assist others in engaging the book, I’ll give introductory versions of three of the stories, representing the attention Hawkins pays to general, number-theoretic, and group-theoretic history. The last two give the “backstory” for three consecutive papers, all published in 1896. These are papers 52, 53, and 54 from the 102 mathematical papers in Frobenius’ collected works. Many other papers from different fields are also given careful attention in Hawkins’ thorough description of the mathematics of Frobenius and its context.

Some general history

The sociology of German mathematics in the 19th and early 20th centuries is intriguing, especially the extreme paucity of what Hawkins calls full professorships. For example, in 1890 Berlin had three such positions and its leading rival Göttingen had two. This environment produced many historically important and somewhat dramatic instances of one famous mathematician being succeeded by another. Hawkins recounts the following events in much greater detail on pages 53–54 and 64–70.

The three full professors in 1890 at Berlin were Fuchs, Kronecker, and Weierstrass, with the latter two no longer on good terms. Weierstrass, then 75, wanted to resign his position, but stayed on because he didn’t want Kronecker to have a lead role in choosing his successor. But the next year Kronecker suddenly died, and Weierstrass was the major force in picking the two new full professors, Schwarz and Frobenius.

Fuchs died in 1902. The committee, including Schwarz and Frobenius, had Hilbert, then one of the two full professors at Göttingen, as their first choice for his successor. Frobenius argued very strenuously that Schottky was a close second. After much back and forth, Göttingen was given a third full professorship filled by Minkowski, Hilbert stayed there, and Schottky became the new full professor at Berlin. The whole incident somewhat sullied Frobenius’ reputation with the hiring authorities: Hawkins calls Frobenius’ arguments for Schottky “specious”, “exaggerated”, and “misleading”.

Schwarz resigned in 1916. This time the committee’s first choice, Schmidt, was chosen as a successor. Again Frobenius highly praised the second choice, Schur, but his word had little weight. After Frobenius died, Schur again came in a close second, this time to Carathéodory. Carathéodory shortly thereafter left for his ancestral homeland of Greece, and Schur was passed over yet again, now in favor of von Mises. Finally, in 1921 Schottky died and Schur, mathematically the natural successor of Frobenius, was finally made full professor.

Many other first-rank mathematicians play important roles in Hawkins’ book. Most prominent among these is Dedekind, eighteen years Frobenius’ senior. Like Frobenius, Dedekind spent some of his early career in Berlin and Zurich. But then he spent most of his career in Brunswick, away from the main centers, never even supervising a doctoral student. Correspondance with Dedekind plays a centrol role in the genesis of the three papers to be discussed next. Hawkins quotes this correspondance extensively, making clear that the two mathematicians had a strong friendship based on deep mutual respect.

Some number-theoretic history

Frobenius had a central role in the early history of what is now called the Chebotarev density theorem. Frobenius’ key paper, number 52 in his collected works, is Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe or On relations between the prime ideals of an algebraic field and the substitutions in its group.

Hawkins describes the very complicated backstory of this paper in pages 318–335. There is a mysterious 15-year publication delay and an incorrect conjecture on a relatively unimportant side issue that made it into the final paper. Hawkins argues that the delay was in part caused by the conjecture.

Frobenius highlights the delay on the first page of his 1896 paper, writing

Ich habe die folgende Arbeit im November 1880 verfasst, und die darin entwickelten Resultate meinen Freunden Stickelberger und Dedekind mitgetheilt.
I have written the following work in November 1880 and communicated the results developed here to my friends Stickelberger and Dedekind.

From Hawkins’ book, we learn a lot more. Frobenius wrote to Dedekind several times, saying that he submitted the work to Crelle’s journal in 1881 and that the review process was taking frustratingly long. Frobenius later wrote again to Dedekind saying that it had finally been accepted. But in fact the paper did not appear. Hawkins speculates that Frobenius withdrew his paper in order to improve it.

In describing the story further, I will use the numbering from Frobenius’ paper. For the benefit of readers who want to pursue Hawkins’ long and careful treatment, I give Hawkins’ numbering as well.

I Theorem 9.14 Equidistribution of factor partitions for polynomials
II Theorem 9.16 Equidistribution of factor partitions for fields
§3 Conjecture 9.17 Converse conjecture
§4 Theorem 9.18 Construction of Frobenius elements
IV Theorem 9.20 Equidistribution of Frobenius divisions in Galois groups
V Conjecture 9.21 Equidistribution of Frobenius conjugacy classes in Galois groups

The equidistribution assertions I, II, IV, and V are similar statements of increasing depth. I will present them here with the technical notion of Dirichlet density replaced by a rough intuitive equivalent, frequency. The material in Sections 3 and 4 of Frobenius’ paper plays a central role but isn’t numbered there.

Frobenius’ Theorem I concerns polynomials and converts a fundamental analytic equality published in 1880 by Kronecker into a group-theoretic statement offering complementary insights. Suppose given an irreducible monic polynomial \(\Phi(x)\) with integral coefficients. Then one can factor it into irreducibles modulo primes. Taking \(\Phi(x) = x^6-6 x^4+6 x^2-6 x+2 \in \mathbb{Z}[x]\) as a running example, its factorizations modulo the first six primes are as in the second column:

\[ \label{firsttable} {\renewcommand{\arraycolsep}{2pt} \begin{array}{ccccc} & & & \mbox{Factor} & \!\!\!\mbox{Factorization} \!\!\! \\ p & \mbox{Factorization of \(\Phi(x)\) in \(\mathbb{F}_p[x]\)} & \mbox{Type} & \mbox{Partition} & \mbox{of \((p)\)} \\ \hline 2 & x^6 & \text{bad} & [1,1,1,1,1,1] & P_1^6 \\ 3 & (x+1)^3 (x+2)^3 & \text{bad} & [1,1,1,1,1,1] & P_{1a}^3 P_{1b}^3\\ 5 & \left(x^2+x+2\right)^2 \left(x^2+3 x+3\right) & \text{bad} & [2,2,2] & P_{2a}^2 P_{2b} \\ 7 & \left(x^3+x^2+5 x+4\right) \left(x^3+6 x^2+4 x+4\right) & \text{good} & [3,3] & P_{3a} P_{3b} \\ 11 &\left(x^4+x^3+10 x^2+3 x+5\right) \left(x^2+10 x+7\right)& \text{good} & [4,2] & P_4 P_2\\ 13 & \left(x^3+6 x^2+10 x+10\right) \left(x^3+7 x^2+7 x+8\right) & \text{good} & [3,3] & P_{3a} P_{3b} \\ \end{array} } \tag{1} \]

Using modern language, say that a prime is bad if a factor is repeated and otherwise good. Always there are only finitely many bad primes and in the example the only bad primes are 2, 3, and 5. The important thing to be extracted from a prime is its associated factor partition, obtained by listing the degrees of the irreducible factors.

To state Frobenius’ Theorem I, one needs to bring in Galois groups. The given polynomial \(\Phi(x)\) has a Galois group \(G\), which is a group of permutations of the polynomial’s complex roots. In the example, the complex roots can be labeled \(r_{12}\), \(r_{13}\), \(r_{14}\), \(r_{23}\), \(r_{24}\), and \(r_{34}\) in such a way that \(G\) becomes identified with the symmetric group \(S_4\). Thus, for example, the transposition \((1,2)\) in \(S_4\) acts as the permutation \((r_{13},r_{23})(r_{14},r_{24})\).

Frobenius’ Theorem I then says that a given partition arises as a factor partition for primes with the same frequency that it arises as a cycle partition for Galois group elements. In the example, the frequencies are as follows:

\[ \label{secondtable} \begin{array}{ccrrrr} \mbox{Partition} & \mbox{# of \(g\)} & p_1 & p_2 & p_3 & \mbox{# of \(p\)} \!\!\! \\ \hline \! [1,1,1,1,1,1] \! & 1 & 157 & 307 & 409 & 968 \\ \! [2,2,1,1] \!& 9 & 17 & 31 & 43 & 9030 \\ \! [2,2,2] \! & 0 & 5 & & & 1 \\ \! [3,3] \! & 8 & 7 & 13 & 19 & 8005 \\ \! [4,2] \! & 6 & 11 & 23 & 29 & 5996 \\ \end{array} \tag{2} \]

So, reading the second row as an example, 9 of the 24 group elements have cycle partition \([2,2,1,1]\), the first three primes having factor partition \([2,2,1,1]\) are \(17\), \(31\), and \(43\), and altogether \(9030\) of the first \(24000\) primes have factor partition \([2,2,1,1]\). The agreement between the “# of \(g\)” and “# of \(p\)” columns is unmistakable. While Frobenius and his contemporaries never saw data as in the last column, we can now easily produce it in under ten seconds.

To state Frobenius’ Theorem II, one needs to translate into the language of prime ideals, a new theory in Frobenius’ time due to Dedekind. Let \(R\) be the ring of integers in the number field \(\mathbb{Q}[x]/f(x)\). Then a prime ideal \((p)\) in \(\mathbb{Z}\) factors into powers of distinct prime ideals in \(R\), as illustrated by the last column in table (1). Numbers in subscripts indicate the degree of a prime ideal, as in \(|R/P_f| = p^f\). By listing degrees of prime ideals, one gets again a partition of the degree of \(\Phi(x)\). At a good prime, the new ideal-theoretic partition is the same as the old polynomial-theoretic partition. At a bad prime, it may be different, although for all three bad primes in the example it is the same. Frobenius’ Theorem II now just repeats Theorem I, except that it refers to ideal-theoretic partitions.

Frobenius points out in Section 3 that if a partition \([f_1,...,f_e]\) doesn’t come from a group element then Theorem II says it can only come from a set of primes of density zero. But a density zero set is not necessarily empty. In Frobenius’ words,

Damit wäre aber nicht ausgeschlossen, dass solche Primzahlen in endlicher und sogar in unendlicher Anzahl existieren.
But it would not be excluded that such primes exist in finite and even infinite number.

Frobenius conjectures in italics that such primes do not exist. He then proves in the rest of Section 3 that the conjecture is true for good primes.

It is surprising that Frobenius didn’t simply ignore bad primes and state his result on good primes as a theorem. After all, bad primes, being finite in number, have no role in the equidistribution statements of the paper. In retrospect, we now know that the conjecture for bad primes is in fact false, as illustrated by [2,2,2] arising from the prime 5 in tables (1) and (2) above.

Also surprising is that Frobenius’ incorrect conjecture went uncorrected for more than 100 years. The first published counterexample to Frobenius’ full conjecture is actually in Hawkins’ book. Hawkins acknowledges Serre as helping in many ways thoughout his forty-year study of Frobenius, and this counterexample is due to Serre. The counterexample I’ve presented above is a modification which has the feature that the polynomial-theoretic factor partitions agree with ideal-theoretic factor partitions.

Having explained some of the mathematics, we can now return to the story. In the early 1880s, Frobenius thought that late 1870s unpublished work of Dedekind would let him prove that if \([f_1,f_2,\dots,f_e]\) doesn’t come from a Galois group element then it never comes from a bad prime either. He was in correspondance with Dedekind on this point. Frobenius seems to have been expecting that Dedekind would publish his recent work, and he would then be able to perfect his statement in Section 3 to include bad primes. Dedekind didn’t publish and Frobenius didn’t publish either.

In the late 1870s, Dedekind had indeed made a fundamental construction \(p \mapsto F_p\), associating to a good prime \(p\) a conjugacy class in \(G\). Section 4 of Frobenius’ paper gives this construction as well and we now call \(F_p\) a Frobenius class. In the 1890s, Hilbert was independently discovering some of Dedekind’s old results, including the construction \(p \mapsto F_p\), and Dedekind was thereby pushed to publish some of his old results. Then Hurwitz was independently discovering Frobenius’ Theorem IV and Frobenius came out with his 1896 paper. Frobenius’ statement V, the most refined and most natural statement about the equidistribution of the \(F_p\), had to wait until the 1920s to be proved by Chebotarev. Remarkable history that one does not know simply by working in present-day algebraic number theory!

Some group-theoretic history

Frobenius was the central figure in the discovery of the theory of representations of finite groups. In pages 449–488, Hawkins describes the backstory behind Frobenius’ first two contributions, Papers 53 and 54. A lesson here is that the historically first route from a group to its character table has a spectacular theoretical simplicity that we have all but forgotten. The concept of group determinant plays the central role, even though it just barely registers on the consciousness of modern mathematicians.

To explain this concept via an example, consider the symmetric group \(S_3\). Abbreviate its elements via \((a,b,c,d,e,f) = (\mbox{Id},(123),(132),(12),(13),(23))\). Then its multiplication table, normalized so that identity elements go down the main diagonal, has the following form:

\[ \begin{array}{c|cccccc} x & a & b & c & d & e & f \\ \hline a & a & b & c & d & e & f \\ c & c & a & b & e & f & d \\ b & b & c & a & f & d & e \\ d & d & e & f & a & b & c \\ e & e & f & d & c & a & b \\ f & f & d & e & b & c & a \end{array}. \]

The group determinant \(\Theta(S_3)\) is just the determinant of this matrix, viewed as a polynomial in \(\mathbb{C}[a,b,c,d,e,f]\). For a general finite group \(G\), its group determinant \(\Theta(G)\) is likewise the determinant of its multiplication table.

Expanded out, \(\Theta(S_3)\) has 147 terms. Factored, however, it takes the following remarkable form:

\[ \begin{eqnarray} \nonumber \Theta(S_3) & = & (a+b+c+d+e+f)(a+b+c-d-e-f) \cdot \\ \label{firsts3} && \qquad \left(a^2-a b-a c+b^2-b c+c^2-d^2+d e+d f-e^2+e f-f^2\right)^2. \end{eqnarray} \tag{3} \]

Dedekind had obtained this formula in 1886, as Hawkins explains on pages 449–450. He knew theoretically that \(\Theta(G)\) factors into linear factors for all abelian \(G\). He had also found factorizations similiar to (3) for other small nonabelian \(G\). He wrote to Frobenius on March 19, 1896, trying to interest him in group determinants.

Frobenius was more than interested and within a month he had fundamental results. For general \(G\) with \(k\) conjugacy classes, he proved a factorization into irreducibles of the form

\[ \Theta(G) = \prod_{i=1}^k \Phi_i^{f_i}. \tag{4} \]

Frobenius sounds particularly modern when he writes about \(k\) being both the number of prime factors and the number of conjugacy classes: he says that this agreement is “all the more remarkable since there does not seem to be any relation between the individual prime factors and the individual classes” (page 476).

Frobenius went further in this first month. The three conjugacy classes of \(S_3\) are \([1,1,1]=\{a\}\), \([3] = \{b,c\}\), and \([2,1] = \{d,e,f\}\). Replacing \(b\) and \(c\) by \(u/2\), and \(d\), \(e\), and \(f\) by \(v/3\), the specialized group determinant factors even further,

\[ \Theta^*(S_3) = \frac{1}{16} (a+u+v) (a+u-v) (2 a-u)^4. \tag{5} \]

For general \(G\), Frobenius passed from group elements to conjugacy classes in the same way. He proved that there is always a factorization into linear factors,

\[ \Theta^*(G) = \frac{1}{\prod_{i=1}^k \deg(\Phi_i)^{\deg(\Phi_i) f_i} }\prod_{i=1}^k \phi_i^{\deg(\Phi_i) f_i}. \tag{6} \]

We recognize now that the entire character table of a general finite group \(G\) has just appeared in (6)! For example, (5) exhibits this character table for \(S_3\):

\[ \begin{array}{c|crr} & \!\![1,1,1]\!\! & [3] \!\! & [2,1] \!\!\!\!\! \\ \hline \phi_1 & 1 & 1 & 1 \\ \phi_2 & 1 & 1 & -1 \\ \phi_3 & 2 & -1 & 0 \end{array} \;\; . \]

Again Frobenius anticipates the modern point of view when he calls (6) “one of the most important formulas” (page 476). In reporting the above results on April 17, 1896, Frobenius writes to Dedekind, “I am very grateful to you for suggesting this work, which has given me immeasurable joy.”

The heroic month just summarized is followed by several equally heroic years, all well described by Hawkins. Immediately, Frobenius replaces the group determinant with computationally practical methods. Already on April 26, 1896 he writes to Dedekind, about “an act of vile ingratitude against the magnificent determinant” \(\Theta\), “the miraculous source from which everything wonderful has flowed.” He is starting to rederive (4) and (6) “directly from the theory of groups” (page 477).

Indeed Frobenius’ first 1896 group theory paper, Über Gruppencharaktere, initiates a general theory of character tables with sophisticated explicit examples, all without group determinants. His other 1896 group theory paper, Über die Primfactoren der Gruppendeterminante, is the high point of group determinants in Frobenius’ published works. In it, (4) and (6) are proved, together with the important supplement that \(f_i = \mbox{deg}(\Phi_i)\). No publication delays here!

Summarizing from a different point of view, group multiplication tables are a required topic in every introductory abstract algebra course, but they offer very little insight as to the nature of a given group \(G\). Character tables are too advanced for such courses, but are an essential part of a good understanding of \(G\). How many instructors know the tight relation between the two types of tables?


The quote from Serre’s preface to Frobenius’ collected works continues very curiously. Analysis of Frobenius’ work and influence would have been very difficult

… et peu utile.
… and of little use.

Serre explains what he means by immediately quoting a letter he received from Brauer: “if the reader wants to get an idea about the importance of Frobenius’ work today, all he has to do is to look at books and papers on groups.” Surely this curious continuation is simply a rhetorical device, meant to underscore the large influence Frobenius has on parts of mathematics to the present day.

Nevertheless, it’s worth saying clearly: Hawkins’ work is extraordinarily useful. It allows the mathematical community, even the great majority of us who do not read German well, to understand the work of the very important mathematician Frobenius. The great length of the book is essential to the book’s success. We are given a sense of the full arc and startling variety of Frobenius career. The careful attention to context is also critical. It gives us a fuller sense of Frobenius’ times and its direct relevance to ours.

1. A Berlin Education

2. Professor at the Zurich Polytechnic

3. Berlin Professor

4. The Paradigm

5. Further Development of the Paradigm

6. The Problem of Pfaff

7. The Cayley-Hermite Problem and Matrix Algebra

8. Arithmetical Investigations: Linear Algebra

9. Arithmetical Investigations: Groups

10. Abelian Functions

11. Frobenius' Generalized Theory of Theta Functions

12. The Group Determinant Problem

13. Group Characters and Representations

14. Alternate Routes to Representation Theory

15. Characters and Representations after 1897

16. Loose Ends

17. Nonnegative Matrices

18. The Mathematics of Frobenius in Retrospect