History of group theory
[1][2][3] Joseph Louis Lagrange, Niels Henrik Abel and Évariste Galois were early researchers in the field of group theory.
However, this work was somewhat isolated, and 1846 publications of Augustin Louis Cauchy and Galois are more commonly referred to as the beginning of group theory.
One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4.
[4] Nicholas Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation,[5] and Thomas Le Seur (1703–1770) (1748)[6][7] and Edward Waring (1762 to 1782) still further elaborated the idea.
Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic.
[8][3][9] Lagrange's goal (1770, 1771) was to understand why equations of third and fourth degree admit formulas for solutions, and a key object was the group of permutations of the roots.
[11] The contemporary work of Alexandre-Théophile Vandermonde (1770) developed the theory of symmetric functions and solution of cyclotomic polynomials.
[13][14] Similarly Cauchy gave credit to both Lagrange and Vandermonde for studying symmetric functions and permutations of variables.
[15][14][better source needed] Paolo Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations.
In succeeding years, Cayley systematically investigated infinite groups and the algebraic properties of matrices, such as the associativity of multiplication, existence of inverses, and characteristic polynomials.
The discontinuous (discrete group) theory was built up by Klein, Lie, Henri Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.
Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem.
[25] Ernst Kummer's attempts to prove Fermat's Last Theorem resulted in work introducing groups describing factorization into prime numbers.
Other group theorists of the 19th century were Joseph Louis François Bertrand, Charles Hermite, Ferdinand Georg Frobenius, Leopold Kronecker, and Émile Mathieu;[3] as well as William Burnside, Leonard Eugene Dickson, Otto Hölder, E. H. Moore, Ludwig Sylow, and Heinrich Martin Weber.
The convergence of the above three sources into a uniform theory started with Jordan's Traité and Walther von Dyck (1882) who first defined a group in the full modern sense.
During the 1880-1920 period, groups described by presentations came into a life of their own through the work of Cayley, Walther von Dyck, Max Dehn, Jakob Nielsen, Otto Schreier, and continued in the 1920-1940 period with the work of H. S. M. Coxeter, Wilhelm Magnus, and others to form the field of combinatorial group theory.
Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu), and in the 1870s Klein's group-theoretic vision of geometry was being realized in his Erlangen program.
The study was continued by Moore and Burnside, and brought into comprehensive textbook form by Leonard Dickson in 1901.
The study was continued by Frank Nelson Cole (up to 660) and Burnside (up to 1092), and finally in an early "millennium project", up to 2001 by Miller and Ling in 1900.
Algebraic groups, defined as solutions of polynomial equations (rather than acting on them, as in the earlier century), benefited heavily from the continuous theory of Lie.
The Burnside problem had tremendous progress, with better counterexamples constructed in the 1960s and early 1980s, but the finishing touches "for all but finitely many" were not completed until the 1990s.
The work on the Burnside problem increased interest in Lie algebras in exponent p, and the methods of Michel Lazard began to see a wider impact, especially in the study of p-groups.
Many 18th and 19th century problems are now revisited in this more general setting, and many questions in the theory of the representations of groups have answers.
Its importance to contemporary mathematics as a whole may be seen from the 2008 Abel Prize, awarded to John Griggs Thompson and Jacques Tits for their contributions to group theory.
