# History of group theory

The

**history of group theory**, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots ofgroup theory : the theory ofalgebraic equation s,number theory andgeometry . [*Harvard citations|nb = yes|last = Wussing|year = 2007*] [*Harvard citations|last = Kleiner|year = 1986|nb = yes*] Harvard citations|last = Smith|year = 1906|nb = yes] Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory.**Early 19th century**The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so 3 important threads in its pre-history are developed here.

**Development of permutation groups**One foundational root of group theory was the quest of solutions of

polynomial equation s of degree higher than 4.An early source occurs in the problem of forming an equation of degree "m" having as its roots "m" of the roots of a given equation of degree "n" > "m". For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and

Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.A common foundation for the theory of equations on the basis of the group of

permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents ("résolvantes, réduites") which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a "Calcul des Combinaisons". The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name "l'assieme delle permutazioni". He also published a letter from Abbati to himself, in which the group idea is prominent.

Galois found that if "r"

_{1}, "r"_{2}, ... "r"_{"n"}are the "n" roots of an equation, there is always a group of permutations of the "r"'s such that

*every function of the roots invariable by the substitutions of the group is rationally known, and

*conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with radicals. Galois also contributed to the theory ofmodular equation s and to that ofelliptic function s. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI). [*Harvard citations|last = Galois|year = 1908|nb = yes*] [*Harvard citations|last = Kleiner|year = 1986|loc = p. 202|nb = yes*] Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now calledGalois theory .Groups similar to Galois groups are (today) called

permutation group s, a concept investigated in particular by Cauchy. A number of important theorems in early group theory is due to Cauchy. Cayley's "On the theory of groups, as depending on the symbolic equation θ^{n}= 1" (1854) gives the first abstract definition offinite group s.**Groups related to geometry**Secondly, the systematic use of groups in geometry, mainly in the guise of

symmetry group s, was initiated by Klein's 1872Erlangen program . [*Harvard citations|last = Wussing|year = 2007|loc = §III.2|nb = yes*] The study of what are now calledLie group s started systematically in 1884 withSophus Lie , followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group ) theory was built up byFelix Klein , Lie, Poincaré, andCharles Émile Picard , in connection in particular withmodular form s andmonodromy .**Appearance of groups in number theory**The third root of group theory was

number theory . Certainabelian group structures had been implicitly used in number-theoretical work by Gauss, and more explicitly byKronecker . [*Harvard citations|last = Kleiner|year = 1986|loc = p. 204|nb = yes*] Early attempts to proveFermat's last theorem were led to a climax by Kummer by introducing groups describing factorization intoprime number s. [*Harvard citations|last = Wussing|year = 2007|loc = §I.3.4|nb = yes*]**Convergence**Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory; by

Camille Jordan , whose "Traité des substitutions et des équations algébriques" (1870 ) is a classic; and toEugen Netto (1882), whose "Theory of Substitutions and its Applications to Algebra" was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand,Charles Hermite , Frobenius,Leopold Kronecker , andEmile Mathieu ; as well as Burnside, Dickson, Hölder, Moore, Sylow, and Weber.The convergence of the above three sources into a uniform theory started with Jordan's "Traité" and von Dyck (

1882 ) who first defined a group in the full modern sense. The textbooks of Weber and Burnside helped establish group theory as a discipline. [*Solomon writes in Burnside's Collected Works, "The effect of [Burnside's book] was broader and more pervasive, influencing the entire course of non-commutative algebra in the twentieth century."*] The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms ofLie algebra s.**Late 19th century**Groups in the 1870-1900 period were described as the continuous groups of Lie, the discontinuous groups, finite groups of substitutions of roots (gradually being called permutations), and finite groups of linear substitutions (usually of finite fields). During the 1880-1920 period, groups described by presentations came into a life of their own through the work of Cayley, von Dyck, Dehn, Nielsen, Schreier, and continued in the 1920-1940 period with the work of Coxeter, Magnus, and others to form the field of

combinatorial group theory .Finite groups in the 1870-1900 period saw such highlights as the

Sylow theorem s, Hölder's classification of groups of square-free order, and the early beginnings of thecharacter theory of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu), and in the 1870sFelix Klein s group-theoretic vision of geometry was being realized in hisErlangen program . The automorphism groups of higher dimensional projective spaces were studied by Jordan in his "Traité" and included composition series for most of the so calledclassical group s, though he avoided non-prime fields and omitted the unitary groups. The study was continued by Moore and Burnside, and brought into comprehensive textbook form by Dickson in 1901. The role ofsimple group s was emphasized by Jordan, and criteria for non-simplicity were developed by Hölder until he was able to classify the simple groups of order less than 200. The study was continued by F. N. 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.Continuous groups in the 1870-1900 period developed rapidly. Killing and Lie's foundational papers were published, Hilbert's theorem in invariant theory 1882, etc.

**Early 20th century**In the period 1900-1940, infinite "discontinuous" (now called

discrete group s) groups gained life of their own. Burnside's famous problem ushered in the study of arbitrary subgroups of finite dimensional linear groups over arbitrary fields, and indeed arbitrary groups.Fundamental group s andreflection group s encouraged the developments ofJ. A. Todd and Coxeter, such as theTodd–Coxeter algorithm in combinatorial group theory.Algebraic group s, defined as solutions of polynomial equations (rather than acting on them, as in the earlier century), benefited heavily from the continuous theory of Lie. Neumann and Neumann produced their study of varieties of groups, groups defined by group theoretic equations rather than polynomial ones.Continuous groups also had explosive growth in the 1900-1940 period. Topological groups began to be studied as such. There were many great achievements in continuous groups: Cartan's classification of semisimple Lie algebras, Weyl's theory of representations of compact groups, Haar's work in the locally compact case.

Finite groups in the 1900-1940 grew immensely. This period witnessed the birth of

character theory by Frobenius, Burnside, and Schur which helped answer many of the 19th century questions in permutation groups, and opened the way to entirely new techniques in abstract finite groups. This period saw the work of Hall: on a generalization of Sylow's theorem to arbitrary sets of primes which revolutionized the study of finite soluble groups, and on the power-commutator structure ofp-group s, including the ideas ofregular p-group s and isoclinism, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Zassenhaus's famousSchur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress onFrobenius group s, and a near classification ofZassenhaus group s.**Mid 20th century**Both depth, breadth and also the impact of group theory subsequently grew. The domain started branching out into areas such as

algebraic group s,group extension s, andrepresentation theory . [*Harvard citations|last = Curtis |year = 2003|nb = yes*] Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify all finitesimple group s in 1982. Completing and simplifying the proof of the classification are areas of active research. [*Harvard citations|last = Aschbacher|year = 2004|nb = yes*]Anatoly Maltsev also made important contributions to group theory during this time; his early work was in logic in the 1930s, but in the 1940s he proved important embedding properties of semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s return to logic proving various theories within the study of groups to be undecidable. Earlier,Alfred Tarski proved elementary group theory undecidable. [*Tarski, Alfred (1953) "Undecidability of the elementary theory of groups" in Tarski, Mostowski, and*]Raphael Robinson "Undecidable Theories". North-Holland: 77-87.**Later 20th century**The period of 1960-1980 was one of excitement in many areas of group theory.

In finite groups, there were many independent milestones. One had the discovery of 22 new sporadic groups, and the completion of the first generation of the

classification of finite simple groups . One had the influential idea of theCarter subgroup , and the subsequent creation of formation theory and the theory of classes of groups. One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field ofcomputational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification.In discrete groups, the geometric methods of Tits and the availability the surjectivity of Lang's map allowed a revolution in algebraic groups. The

Burnside problem had tremendous progress, with better counterexamples constructed in the 60s and early 80s, but the finishing touches "for all but finitely many" were not completed until the 90s. The work on the Burnside problem increased interest in Lie algebras in exponent "p", and the methods of Lazard began to see a wider impact, especially in the study of "p"-groups.Continuous groups broadened considerably, with "p"-adic analytic questions becoming important. Many conjectures were made during this time, including the coclass conjectures.

**Late 20th century**The last twenty years of the twentieth century enjoyed the successes of over one hundred years of study in group theory.

In finite groups, post classification results included the O'Nan Scott theorem, the Aschbacher classification, the classification of multiply transitive finite groups, the determination of the maximal subgroups of the simple groups and the corresponding classifications of

primitive group s. In finite geometry and combinatorics, many problems could now be settled. The modular representation theory entered a new era as the techniques of the classification were axiomatized, including fusion systems, Puig's theory of pairs and nilpotent blocks. The theory of finite soluble groups was likewise transformed by the influential book of Doerk–Hawkes which brought the theory of projectors and injectors to a wider audience.In discrete groups, several areas of geometry came together to produce exciting new fields. Work on

knot theory ,orbifold s,hyperbolic manifold s, and groups acting on trees (theBass–Serre theory ), much enlivened the study ofhyperbolic group s,automatic group s. Questions such as Thurston's 1982geometrization conjecture , inspired entirely new techniques ingeometric group theory andlow dimensional topology , and was involved in the solution of one of theMillennium Prize Problems , thePoincaré conjecture .Continuous groups saw the solution of the problem of

hearing the shape of a drum in 1992 using symmetry groups of thelaplacian operator . Continuous techniques were applied to many aspects of group theory usingfunction space s andquantum group s. 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.**Today**Group theory continues to be a intensely studied matter. Its importance to contemporary mathematics as a whole may be seen from the 2008

Abel Prize , awarded toJohn Griggs Thompson andJacques Tits for their contributions to group theory.**Notes****References*** Historically important publications in group theory.

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* | year=1986 | journal=Mathematics Magazine | issn=0025-570X | volume=59 | issue=4 | pages=195–215

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