History of group theory
Encyclopedia
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 of group theory
: the theory of algebraic equations, number theory
and geometry
. Lagrange
, Abel
and Galois
were early researchers in the field of group theory.
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.
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 r1, r2, ... rn are the n roots of an equation, there is always a group of permutations of the rs such that
In modern terms, the solvability
of the Galois group attached to the equation determines the solvability of the equation with radicals.
Galois is the first to use the words group (groupe in French) and primitive in their modern meanings. He did not use primitive group but called equation primitive an equation whose Galois group is primitive. He has discovered the notion of normal subgroup
s and has found that a solvable primitive group may be identified to a subgroup of the affine group
of an affine space
over a finite field
of prime order.
Galois also contributed to the theory of modular equation
s and to that of elliptic 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). Galois is honored as the first mathematician linking group theory and field theory
, with the theory that is now called Galois 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 θn = 1 (1854) gives the first abstract definition of finite group
s.
Secondly, the systematic use of groups in geometry, mainly in the guise of symmetry group
s, was initiated by Klein's
1872 Erlangen program
. The study of what are now called Lie group
s started systematically in 1884 with Sophus Lie
, followed by work of Killing
, Study
, Schur
, Maurer
, and Cartan
. The discontinuous (discrete group
) theory was built up by Felix Klein
, Lie, Poincaré
, and Charles Émile Picard
, in connection in particular with modular form
s and monodromy
.
The third root of group theory was number theory
. Certain abelian group
structures had been implicitly used in number-theoretical
work by Gauss
, and more explicitly by Kronecker. Early attempts to prove Fermat's last theorem
were led to a climax by Kummer
by introducing groups describing factorization into prime number
s.
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 to Eugen 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
, and Émile 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. The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms of Lie algebra
s.
, Walther 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 the character theory
of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu
), and in the 1870s Felix Klein
's group-theoretic vision of geometry was being realized in his Erlangen 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 called classical group
s, though he avoided non-prime fields and omitted the unitary group
s. The study was continued by Moore and Burnside
, and brought into comprehensive textbook form by Leonard Dickson in 1901. The role of simple 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.
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 and reflection group
s encouraged the developments of J. A. Todd
and Coxeter, such as the Todd–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 group
s, 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 of p-group
s, including the ideas of regular p-group
s and isoclinism of groups
, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Zassenhaus
's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress on Frobenius group
s, and a near classification of Zassenhaus group
s.
s, group extension
s, and representation theory
. Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify
all finite simple group
s in 1982. Completing and simplifying the proof of the classification are areas of active research.
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
.
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 the Carter 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 of computational 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.
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 groups. 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 (the Bass–Serre theory
), much enlivened the study of hyperbolic group
s, automatic group
s. Questions such as Thurston
's 1982 geometrization conjecture
, inspired entirely new techniques in geometric group theory
and low dimensional topology, and was involved in the solution of one of the Millennium Prize Problems
, the Poincaré conjecture
.
Continuous groups saw the solution of the problem of hearing the shape of a drum
in 1992 using symmetry groups of the laplacian operator. Continuous techniques were applied to many aspects of group theory using function space
s and quantum 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.
, awarded to John Griggs Thompson and Jacques Tits
for their contributions to group theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
domain studying groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in their various forms, has evolved in various parallel threads. There are three historical roots of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
: the theory of algebraic equations, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
, Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
and Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
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 LagrangeJoseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
and Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
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 equations 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
Johann van Waveren Hudde
Johannes Hudde was a burgomaster of Amsterdam between 1672 – 1703, a mathematician and governor of the Dutch East India Company....
(1659). Saunderson
Nicholas Saunderson
Nicholas Saunderson was an English scientist and mathematician. According to one leading historian of statistics, he may have been the earliest discoverer of Bayes theorem.-Biography:...
(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
Edward Waring
Edward Waring was an English mathematician who was born in Old Heath , Shropshire, England and died in Pontesbury, Shropshire, England. He entered Magdalene College, Cambridge as a sizar and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of...
(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
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
(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
Alexandre-Théophile Vandermonde
Alexandre-Théophile Vandermonde was a French musician, mathematician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.Vandermonde was a violinist, and became engaged with...
(1770) also foreshadowed the coming theory.
Ruffini
Paolo Ruffini
Paolo Ruffini was an Italian mathematician and philosopher.By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics...
(1799) attempted a proof of the impossibility of solving the quintic
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
and higher equations. Ruffini distinguished what are now called intransitive and transitive
Transitivity (mathematics)
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
, and imprimitive
Primitive permutation group
In mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X...
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.
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
found that if r1, r2, ... rn are the n roots of an equation, there is always a group of permutations of the rs 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
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
of the Galois group attached to the equation determines the solvability of the equation with radicals.
Galois is the first to use the words group (groupe in French) and primitive in their modern meanings. He did not use primitive group but called equation primitive an equation whose Galois group is primitive. He has discovered the notion of normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s and has found that a solvable primitive group may be identified to a subgroup of the affine group
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
of an affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
of prime order.
Galois also contributed to the theory of modular equation
Modular equation
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.The most frequent use of the term...
s and to that of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
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). Galois is honored as the first mathematician linking group theory and field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, with the theory that is now called Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
.
Groups similar to Galois groups are (today) called permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
s, a concept investigated in particular by Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
. A number of important theorems in early group theory is due to Cauchy. Cayley's
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s.
Groups related to geometry
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s, was initiated by Klein's
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
1872 Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...
. The study of what are now called Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s started systematically in 1884 with Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
, followed by work of Killing
Wilhelm Killing
Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
, Study
Eduard Study
Eduard Study was a German mathematician known for work on invariant theory of ternary forms and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.Study was born in Coburg in the Duchy of...
, Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
, Maurer
Ludwig Maurer
Ludwig Maurer was a German mathematician professor on the Tübingen University. He was the eldest son of Konrad von Maurer and Valerie von Faulhaber . His 1887 dissertation at the University of Strassburg was on the theory of linear substitutions, known today as matrix groups...
, and Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
. The discontinuous (discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
) theory was built up by Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, Lie, Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, and Charles Émile Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
, in connection in particular with modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s and monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
.
Appearance of groups in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Certain abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
structures had been implicitly used in number-theoretical
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
work by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, and more explicitly by Kronecker. Early attempts to prove Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
were led to a climax by Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
by introducing groups describing factorization into prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s.
Convergence
Joseph Alfred Serret
Joseph Alfred Serret was a French mathematician who was born in Paris, France, and died in Versailles, France.-Books by J. A. Serret:* * *...
, who devoted section IV of his algebra to the theory; by Camille Jordan
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
, whose Traité des substitutions et des équations algébriques (1870) is a classic; and to Eugen Netto
Eugen Netto
Eugen Otto Erwin Netto was a German mathematician. He was born in Halle and died in Giessen. He is listed in the at the Clark University website. A short is available at the MacTutor History of Mathematics archive....
(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
Joseph Louis François Bertrand
Joseph Louis François Bertrand was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics....
, Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....
, Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
, and Émile Mathieu
Émile Léonard Mathieu
Émile Léonard Mathieu was a French mathematician. He is most famous for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathieu transformation...
; as well as Burnside
William Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
, Dickson
Leonard Eugene Dickson
Leonard Eugene Dickson was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory.-Life:Dickson considered himself a Texan by...
, Hölder
Otto Hölder
Otto Ludwig Hölder was a German mathematician born in Stuttgart.Hölder first studied at the Polytechnikum and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstraß, and Ernst Kummer.He is famous for many things including: Hölder's inequality, the Jordan–Hölder...
, Moore
E. H. Moore
Eliakim Hastings Moore was an American mathematician.-Life:Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, discovered mathematics through a summer job at the Cincinnati Observatory while in high school. He learned mathematics at Yale University, where he was...
, Sylow
Peter Ludwig Mejdell Sylow
Peter Ludwig Mejdell Sylow was a Norwegianmathematician, who proved foundational results in group theory. He was born and died in Christiania ....
, and Weber.
The convergence of the above three sources into a uniform theory started with Jordan's
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
Traité and von Dyck
Walther von Dyck
Walther Franz Anton von Dyck , born Dyck and later ennobled, was a German mathematician...
(1882) who first defined a group in the full modern sense. The textbooks of Weber and Burnside helped establish group theory as a discipline. The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms of Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
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 Arthur CayleyArthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, Walther von Dyck
Walther von Dyck
Walther Franz Anton von Dyck , born Dyck and later ennobled, was a German mathematician...
, Dehn
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
, Nielsen
Jakob Nielsen (mathematician)
Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium...
, Schreier
Otto Schreier
Otto Schreier was an Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. He studied mathematics at the University of Vienna and obtained his doctorate in 1923, under the supervision of Philipp Furtwängler...
, and continued in the 1920-1940 period with the work of Coxeter
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
, Magnus
Wilhelm Magnus
Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....
, and others to form the field of combinatorial group theory
Combinatorial group theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations...
.
Finite groups in the 1870-1900 period saw such highlights as the Sylow theorem
Sylow theorem
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains...
s, Hölder
Otto Hölder
Otto Ludwig Hölder was a German mathematician born in Stuttgart.Hölder first studied at the Polytechnikum and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstraß, and Ernst Kummer.He is famous for many things including: Hölder's inequality, the Jordan–Hölder...
's classification of groups of square-free order, and the early beginnings of the character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu
Mathieu
Mathieu may refer to:In places:* Mathieu, Calvados, a commune of the Calvados département in FranceIn mathematics:* Claude-Louis Mathieu , French mathematician and astronomer...
), and in the 1870s Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's group-theoretic vision of geometry was being realized in his Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...
. The automorphism groups of higher dimensional projective spaces were studied by Jordan in his Traité and included composition series for most of the so called classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s, though he avoided non-prime fields and omitted the unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
s. The study was continued by Moore and Burnside
Burnside
- Places :Australia* City of Burnside, a Local Government Area of Adelaide, South Australia* Burnside, South Australia, a suburb of the City of Burnside* Burnside, Victoria, a suburb of MelbourneCanada* Burnside Business Park, in Dartmouth, Nova Scotia...
, and brought into comprehensive textbook form by Leonard Dickson in 1901. The role of simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
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
Frank Nelson Cole
Frank Nelson Cole, Ph.D. was an American mathematician, born in Ashland, Massachusetts, and educated at Harvard, where he lectured on mathematics from 1885 to 1887....
(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 groupDiscrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
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
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s and reflection group
Reflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a...
s encouraged the developments of J. A. Todd
J. A. Todd
John Arthur Todd FRS was a British geometer. He was born in Liverpool, and went to Trinity College of the University of Cambridge in 1925. He did research under H.F. Baker, and in 1931 took a position at the University of Manchester. He became a lecturer at Cambridge in 1937...
and Coxeter, such as the Todd–Coxeter algorithm in combinatorial group theory. Algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
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
Hanna Neumann
Johanna Neumann was a German-born mathematician who worked on group theory.Johanna was born in Lankwitz, Steglitz-Zehlendorf, Germany. She attended Auguste-Viktoria-Schule and the University of Berlin and completed her studies in 1936 with distinctions in mathematics and physics. She began...
produced their study of varieties of group
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
s, 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
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
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
Philip Hall
Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
: 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 of p-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
s, including the ideas of regular p-group
Regular p-group
In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian p-groups, but is general enough to include most "small" p-groups...
s and isoclinism of groups
Isoclinism of groups
In mathematics, specifically group theory, isoclinism is an equivalence relation on groups that is broader than isomorphism, that is, any two groups that are isomorphic are isoclinic, but two isoclinic groups may not be isomorphic. The concept of isoclinism was introduced by to help classify and...
, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Zassenhaus
Hans Julius Zassenhaus
Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra....
's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress on Frobenius group
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
s, and a near classification of Zassenhaus group
Zassenhaus group
In mathematics, a Zassenhaus group, named after Hans Julius Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.- Definition :...
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 groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, group extension
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
s, and representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
. Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
all finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s in 1982. Completing and simplifying the proof of the classification are areas of active research.
Anatoly Maltsev
Anatoly Maltsev
Anatoly Ivanovich Maltsev was born in Misheronsky, near Moscow, and died in Novosibirsk, USSR. He was a mathematician noted for his work on the decidability of various algebraic groups...
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
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
proved elementary group theory undecidable
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...
.
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
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. One had the influential idea of the Carter subgroup
Carter subgroup
In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a subgroup H that is a nilpotent group, and self-normalizing...
, 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 of computational group theory
Computational group theory
In mathematics, computational group theory is the study ofgroups by means of computers. It is concernedwith designing and analysing algorithms anddata structures to compute information about groups...
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
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
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 groups. 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
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
s, hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
s, and groups acting on trees (the Bass–Serre theory
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
), much enlivened the study of hyperbolic group
Hyperbolic group
In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...
s, automatic group
Automatic group
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.More...
s. Questions such as Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
's 1982 geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
, inspired entirely new techniques in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and low dimensional topology, and was involved in the solution of one of the Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
, the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
.
Continuous groups saw the solution of the problem of hearing the shape of a drum
Hearing the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...
in 1992 using symmetry groups of the laplacian operator. Continuous techniques were applied to many aspects of group theory using function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s and quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
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 an intensely studied matter. Its importance to contemporary mathematics as a whole may be seen from the 2008 Abel PrizeAbel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
, awarded to John Griggs Thompson and Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
for their contributions to group theory.