VI.58 Ferdinand Georg Frobenius
b. Berlin, 1849; d. Berlin, 1917
Analysis; linear algebra; number theory; theory of groups;
character theory
After school in Berlin, Frobenius (who suppressed his first name and wrote mainly as G. Frobenius) spent one semester studying mathematics and physics in Göttingen, then returned to Berlin where he studied under KRONECKER [VI.48], KUMMER [VI.40], WEIERSTRASS [VI.44], and others. He wrote his doctoral dissertation (in Latin) supervised by Weierstrass, in 1870, on infinite series representations of analytic functions of one variable. For four years he worked as a schoolteacher in Berlin before he became Außerordentlicher Professor (associate professor) at Berlin University. After less than two years, in 1875, he was called to a full professorship at the Eidgenössische Technische Hochschule in Zürich, where he remained until 1892, when he returned to Berlin as successor to KRONECKER [VI.48]. He retired in 1916, and died one year later.
His early contributions were to analysis and the theory of differential equations. Later he wrote mainly on theta functions, algebra, and number theory. One of his well-known contributions lies across group theory and number theory. Given a polynomial with coefficients in an algebraic number field, one may ask for the degrees of the irreducible factors that occur when it is reduced modulo a prime ideal. In particular, one may ask for the “density” (suitably defined) of the set of prime ideals modulo which a given pattern of irreducible-factor degrees arises. Pursuing ideas of Kronecker, Frobenius proved that, if the GALOIS GROUP [V.21] is the SYMMETRIC GROUP [III.68], then that density is the proportion of elements of the group whose cycle structure is the pattern of degrees. He conjectured that this should be true whatever the Galois group. A tool he used for this led to the name “Frobenius automorphism” for the natural generator a 
aq of the Galois group of a finite extension of the field 
q. The conjecture was proved by N. G. Chebotaryov in 1925 and is now known as the Chebotaryov density theorem, or, sometimes, the Frobenius-Chebotaryov density theorem.
Another well-known and important contribution was to the theory of matrices and linear transformations, where Frobenius introduced the minimal polynomial and other invariants (the elementary divisors).
Frobenius is best known for his work in finite group theory. Like Otto Holder and WILLIAM BURNSIDE [VI.60], he focused for a time on the search for FINITE SIMPLE GROUPS [V.7]. His greatest contribution, however, is his invention of the theory of GROUP CHARACTERS [IV.9]. This emerged unexpectedly in 1896 out of his study of group determinants. These are the determinants of square matrices with rows and columns indexed by the members of a finite group G, and with (a, b)-entry xab-1, where the xg are independent variables, one for each element g of G. His interest, stimulated by correspondence with DEDEKIND [VI.50], was in how the group determinant factorizes as a polynomial in these variables. This problem led Frobenius to the discovery of certain sets of complex numbers, which he called Gruppencharactere, one for each conjugacy class in the group, that arose as the solutions of sets of linear equations connected with the group. Nowadays they are defined differently: for each complex linear representation ρ of the group G (that is, homomorphism ρ : G → GLn(
), where GLn () is the group of n × n invertible matrices over ), the associated character χ is the map G → such that χ(g) = trace ρ(g) for g ∈ G. Frobenius proved the orthogonality relations, recognized the connection of his characters with matrix representations of the group, calculated the character tables of the symmetric groups, the alternating groups, and the Mathieu groups, and used properties of induced characters to prove his famous theorem that a transitive permutation group in which no element other than the identity fixes two or more points has a regular normal subgroup (that is, a subgroup consisting of the identity together with the fixed-point-free elements of the group). To this day no purely group-theoretic proof of this theorem has been found. In recognition of his contribution such groups are now called Frobenius groups. Through character theory and representation theory, as developed by Frobenius for finite groups (and by his pupil, friend, and colleague Issai Schur for classical matrix groups), group theory found important applications in physics and chemistry a generation later.
Further Reading
Begehr, H., ed. 1998. Mathematik in Berlin: Geschichte und Dokumentation, two volumes. Aachen: Shaker.
Curtis, C. W. 1999. Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. Providence, RI: American Mathematical Society.
Serre, J.-P., ed. 1968. F. G. Frobenius: Gesammelte Abhandlungen, three volumes. Berlin: Springer.