View source for History of group theory

The '''history of group theory''', a [[mathematics|mathematical]] domain studying [[group (mathematics)|groups]] in their various forms, has evolved in various parallel threads. There are three historical roots of [[group theory]]: the theory of [[algebraic equation]]s, [[number theory]] and [[geometry]].<ref>{{harvnb|Wussing|2007}}</ref><ref>{{harvnb|Kleiner|1986}}</ref><ref name=Smith>{{harvnb|Smith|1906}}</ref> [[Joseph Louis Lagrange]], [[Niels Henrik Abel]] and [[Évariste Galois]] were early researchers in the field of group theory.
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include summary of the post 1870 work here -->

==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 [[Augustin Louis Cauchy]] and Galois are more commonly referred to as the beginning of group theory.  The theory did not develop in a vacuum, and so three 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 of a polynomial|degree]] higher than 4.

An early source occurs in the problem of forming an equation of degree ''m'' having as its [[root of a polynomial|roots]] ''m'' of the roots of a given equation of degree <math>n > m</math>. For simple cases, the problem goes back to [[Johann van Waveren Hudde]] (1659).<ref>Hudde, Johannes (1659) "Epistola prima, de reductione æquationum" (First letter:  on the reduction of equations). In: Descartes, René; Beaune, Florimond de; Schooten, Frans van; Hudde, Johannes; Heuraet, Hendrik van. [https://babel.hathitrust.org/cgi/pt?id=ucm.5320271924;view=1up;seq=428 ''Renati Des-Cartes Geometria'']. 2nd ed.  vol. 1.  (in Latin) Amsterdam, Netherlands:  Louis and Daniel Elzevir. pp. 406–506.</ref> [[Nicholas Saunderson]] (1740) noted that the determination of the [[quadratic polynomial|quadratic]] factors of a biquadratic expression necessarily leads to a [[sextic equation]],<ref>{{cite book |last1=Saunderson |first1=Nicholas |title=The Elements of Algebra, in Ten Books |date=1740 |publisher=Cambridge University Press |location=Cambridge, England |volume=2 |pages=735–736, "Of the resolution of all sorts of biquadratic equations by the mediation of cubics." |url=https://books.google.com/books?id=1NI_AQAAMAAJ&pg=PA735}}</ref> and Thomas Le Seur (1703–1770) (1748)<ref>{{cite book |last1=Le Seur |first1=Thomas |title=Memoire sur le Calcul Integral |date=1748 |publisher=Freres Pagliarini |location=Rome, (Italy) |url=https://archive.org/details/bub_gb_xAQfNL3OiHMC |language=French}} ; pp. 13 ff, see especially pp. 22–23.</ref><ref>Articles about Thomas Le Seur are available in [[:fr:Thomas Leseur|French Wikipedia]] and [[:de:Thomas Le Seur|German Wikipedia]].</ref> and [[Edward Waring]] (1762 to 1782) still further elaborated the idea. Waring [[mathematical proof|proved]] the [[fundamental theorem of symmetric polynomials]], and specially considered the relation between the roots of a [[quartic equation]] and its resolvent cubic.<ref>See:
* {{cite book |last1=Waring |first1=Edward |title=Miscellanea Analytica, de aequationibus algebraicis, et curvarum proprietatibus |date=1762 |publisher=J. Bentham |location=Cambridge, England |url=https://archive.org/details/miscellaneaanal00warigoog/page/n6 |language=Latin}}
* {{cite book |last1=Waring |first1=Edward |title=Meditationes Algebraicæ |date=1770 |publisher=J. Archdeacon |location=Cambridge, England |language=Latin }}
* {{cite book |last1=Waring |first1=Edward |title=Meditationes Algebraicæ |date=1782 |publisher=J. Archdeacon |location=Cambridge, England |edition=3rd |url=https://archive.org/details/bub_gb_1MNbAAAAQAAJ |language=Latin }}</ref><ref name=Smith/><ref>{{cite journal |last1=Burkhardt |first1=Heinrich |title=Die Anfänge der Gruppentheorie und Paolo Ruffini |journal=Zeitschrift für Mathematik und Physik |date=1892 |volume=37 (Supplement) |pages=119–159 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102937661;view=1up;seq=561 |trans-title=The beginnings of group theory and Paolo Ruffini |language=German}}</ref>

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 [[permutation]]s of the roots. On this was built the theory of substitutions.<ref>See:
*  {{cite journal |last1=Lagrange |title=Reflexions sur la résolution algébrique des équations |journal=Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres (Berlin) |date=1770 |volume=1 |pages=134–215 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433009864541;view=1up;seq=224 |trans-title=Reflections on the algebraic solution of equations |language=French}}
*  {{cite journal |last1=Lagrange |title=Suite des reflexions sur la résolution algébrique des équations |journal=Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres (Berlin) |date=1771 |volume=2 |pages=138–253 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433009864558;view=1up;seq=206 |trans-title=Continuation of reflections on the algebraic solution of equations |language=French}}</ref> He discovered that the roots of all [[Lagrange 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''.<ref>{{harvnb|Lagrange|1771|p=235}}</ref> The contemporary work of [[Alexandre-Théophile Vandermonde]] (1770) developed the theory of [[symmetric function]]s and solution of [[cyclotomic polynomial]]s.<ref name=Smith/><ref>{{cite journal |last1=Vandermonde |title=Mémoire sur la resolution des équations |journal=Histoire de l'Académie Royale des Sciences. Avec les Mémoires de Mathématique & de Physique |date=1771 |pages=365–416 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015013757649;view=1up;seq=551 |trans-title=Memoir on the solution of equations |language=French}}</ref> [[Leopold Kronecker]] has been quoted as saying that a new boom in algebra began with Vandermonde's first paper.<ref>{{cite book |last1=Vandermonde |first1=N. |title=Abhandlungen aus der reinen Mathematik |date=1888 |publisher=Julius Springer |url=https://books.google.com/books?id=iKg_AQAAIAAJ&pg=PP11 |language=de|editor-first=Carl|editor-last=Itzigsohn|quote=Mit Vandermonde's im Jahre 1770 der Pariser Akademie vorgelegten Abhand- lung über die Auflösung der Gleichungen beginnt – so hat sich jüngst Herr Kronecker in einer Vorlesung geäussert – der neue Aufschwung der Algebra|trans-quote=With Vandermonde's treatise on the solution of equations presented to the Paris Academy in 1770 – as Kronecker recently said in a lecture – the new boom in algebra begins}}</ref><ref name="VBio"/> Similarly Cauchy gave credit to both Lagrange and Vandermonde for studying symmetric functions and permutations of variables.<ref>{{cite web |last1=Cauchy |first1=A. L. |translator-last=Bertrand |translator-first=Mike |translator-last2=Gaschignard |translator-first2=Stephen |title=Memoire Sur le Nombre des Valeurs|trans-title=Paper on the number of values |url=http://nonagon.org/ExLibris/cauchys-memoire-sur-le-nombre-des-valeurs |website=Ex Libris |date=3 December 2014|orig-date=January 1815}}</ref><ref name="VBio">{{MacTutor|id=Vandermonde|title=Alexandre-Théophile Vandermonde|quote=Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory.}}</ref>{{Better source needed|reason=The first source gives Vandermonde equal priority, while the second states Vandermonde contributed more than Lagrange.|date=August 2022}}

[[Paolo Ruffini (mathematician)|Paolo Ruffini]] (1799) attempted a proof of the impossibility of solving the [[quintic equation|quintic]] and higher equations.<ref>{{cite book |last1=Ruffini |first1=Paolo |title=Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto |trans-title=General Theory of Equations, in which the algebraic solution of general equations of degree higher than four is proven impossible |date=1799 |publisher=St. Tommaso d'Aquino |location=Bologna, (Italy) |volume=1 & 2 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015065507694;view=1up;seq=7 |language=Italian}}</ref> Ruffini was the first person to explore ideas in the theory of [[permutation group]]s such as the [[order (group theory)|order]] of an element of a group, [[conjugacy class|conjugacy]], and the [[cycle decomposition (group theory)|cycle decomposition]] of elements of permutation groups. Ruffini distinguished what are now called intransitive and [[Group action#Types of actions|transitive]], and imprimitive and [[primitive permutation group|primitive]] groups, and (1801) uses the group of an equation under the name ''l'assieme delle permutazioni''. He also published a letter from [[Pietro Abbati]] to himself, in which the group idea is prominent.<ref>{{cite journal |last1=Abbati |first1=Pietro |title=Lettera di Pietro Abbati Modenese al socio Paolo Ruffini |journal=Memorie di Matematica e di Fisica della Società Italiana delle Scienze |date=1803 |volume=10 (part 2) |pages=385–409 |url=https://www.biodiversitylibrary.org/item/34169#page/7/mode/1up |trans-title=Letter from Pietro Abbati of Modena to his colleague Paolo Ruffini |language=Italian}}</ref><ref name=Smith/> However, he never formalized the concept of a group, or even of a permutation group.

[[Image:Evariste galois.jpg|right|thumb|150px|Galois age fifteen, drawn by a classmate.]]
[[Évariste Galois]] is honored as the first mathematician linking group theory and [[field theory (mathematics)|field theory]], with the theory that is now called [[Galois theory]].<ref name=Smith/> Galois also contributed to the theory of [[modular equation]]s and to that of [[elliptic function]]s.<ref>{{harvnb|Galois|1908}}</ref><ref>{{harvnb|Kleiner|1986|p=202}}</ref> His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the posthumous publication of his collected papers in 1846 (Liouville, Vol. XI). He considered for the first time what is now called the [[closure (mathematics)|closure]] property of a group of permutations, which he expressed as
{{quote|if in such a group one has the substitutions S and T then one has the substitution ST.}}
Galois found that if <math>r_1, r_2, \ldots, r_n</math> 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 [[solvable group|solvability]] of the [[Galois group]] attached to the equation determines the [[solvable by radicals|solvability]] of the equation with radicals.

Galois was 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 group|primitive]]. He discovered the notion of [[normal subgroup|normal]] [[subgroup]]s and 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 number|prime]] order.<ref>{{cite web |title=Galois' last letter |url=http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament}}</ref>

Groups similar to Galois groups are (today) called [[permutation group]]s. The theory of permutation groups received further far-reaching development in the hands of [[Augustin Cauchy]] and [[Camille Jordan]], both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of [[group isomorphism|isomorphism]], although limited to the context of permutation groups. It was also Jordan who put the term ''group'' in wide use.

An ''abstract'' notion of a ([[finite group|finite]]) group appeared for the first time in [[Arthur Cayley]]'s 1854 paper ''On the theory of groups, as depending on the symbolic equation <math>\theta^n = 1</math>''.<ref>{{cite journal |last1=Cayley |first1=A. |title=On the theory of groups, as depending on the symbolic equation θ<sup>n</sup> = 1 |journal=Philosophical Magazine |date=1854 |volume=7 |issue=42 |pages=40–47 |url=https://babel.hathitrust.org/cgi/pt?id=pst.000068485757;view=1up;seq=54 |series=4th series |doi=10.1080/14786445408647421}}</ref><ref>{{MacTutor|class=HistTopics|id=Abstract_groups|title=The abstract group concept}}</ref> Cayley proposed that any finite group is isomorphic to a subgroup of a permutation group, a result known today as [[Cayley's theorem]]. In succeeding years, Cayley systematically investigated infinite groups and the algebraic properties of [[matrix (mathematics)|matrices]], such as the [[associativity]] of multiplication, existence of [[inverse matrix|inverses]], and [[characteristic polynomial]]s.

===Groups related to geometry===
[[Image:Felix Klein.jpeg|right|thumb|150px|Felix Klein]]
[[Image:Sophus Lie.jpg|left|thumb|150px|Sophus Lie]]
Secondly, the systematic use of groups in geometry, mainly in the guise of [[symmetry group]]s, was initiated by [[Felix Klein]]'s 1872 [[Erlangen program]].<ref>See:
*  {{cite book |last1=Klein |first1=Felix |title=Vergleichende Betrachtungen über neuere geometrische Forschungen |trans-title=A comparative review of recent researches in geometry |date=1872 |publisher=Andreas Deichert |location=Erlangen, Germany |url=https://archive.org/details/vergleichendebe00kleigoog |language=German}}
*  Reprinted in:  {{cite journal |last1=Klein |first1=Felix |title=Vergleichende Betrachtungen über neuere geometrische Forschungen |journal=Mathematische Annalen |date=1892 |volume=43 |issue=1 |pages=63–100 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.b3792844;view=1up;seq=73 |trans-title=A comparative review of recent researches in geometry |language=German |doi=10.1007/bf01446615|s2cid=60620433 }}
*  English translation:  {{cite book<!--Deny Citation Bot--> |last=Klein |first=Felix C. |translator-last=Haskell |translator-first=M.W. |editor-last=Rughoonauth |editor-first=N.C. |orig-year=1892 |year=2008 |title=A comparative review of recent researches in geometry |arxiv=0807.3161}}</ref><ref>{{harvnb|Wussing|2007|loc = §III.2}}</ref> The study of what are now called [[Lie group]]s started systematically in 1884 with [[Sophus Lie]], followed by work of [[Wilhelm Killing]], [[Eduard Study]], [[Issai Schur]], [[Ludwig Maurer]], and [[Élie Cartan]]. The discontinuous ([[discrete group]]) theory was built up by Klein, Lie, [[Henri Poincaré]], and [[Charles Émile Picard]], in connection in particular with [[modular form]]s and [[monodromy]].

===Appearance of groups in number theory===
[[Image:ErnstKummer.jpg|right|thumb|150px|Ernst Kummer]]
The third root of group theory was [[number theory]]. [[Leonhard Euler]] considered [[algebraic operation]]s on numbers modulo an integer—[[modular arithmetic]]—in [[Euler's theorem|his generalization]] of [[Fermat's little theorem]]. These investigations were taken much further by [[Carl Friedrich Gauss]], who considered the structure of [[multiplicative group of integers modulo n|multiplicative groups of residues mod ''n'']] and established many properties of [[cyclic group|cyclic]] and more general [[abelian group|abelian]] groups that arise in this way. In his investigations of [[composition of binary quadratic forms]], Gauss explicitly stated the associative law for the composition of forms. In 1870, [[Leopold Kronecker]] gave a definition of an abelian group in the context of [[ideal class group]]s of a [[number field]], generalizing Gauss's work.<ref>{{harvnb|Kleiner|1986|loc = p. 204}}</ref> [[Ernst Kummer]]'s attempts to prove [[Fermat's Last Theorem]] resulted in work introducing [[class group|groups describing factorization]] into prime numbers.<ref>{{harvnb|Wussing|2007|loc = §I.3.4}}</ref> In 1882, [[Heinrich M. Weber]] realized the connection between permutation groups and abelian groups and gave a definition that included a two-sided [[cancellation property]] but omitted the existence of the [[inverse element]], which was sufficient in his context (finite groups).{{sfn|Kleiner|2007|p=32}}

===Convergence===
[[Image:Camille Jordan 4.jpg|left|thumb|150px|Camille Jordan]]
Group theory as an increasingly independent subject was popularized by [[Joseph Alfred Serret|Serret]], who devoted section IV of his algebra to the theory; by [[Camille Jordan]], whose ''[[List of important publications in mathematics#Traité des substitutions et des équations algébriques|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 19th century were [[Joseph Louis François Bertrand]], [[Charles Hermite]], [[Ferdinand Georg Frobenius]], [[Leopold Kronecker]], and [[Émile Léonard Mathieu|Émile Mathieu]];<ref name=Smith/> as well as [[William Burnside]], [[Leonard Eugene Dickson]], [[Otto Hölder]], [[E. H. Moore]], [[Peter Ludwig Mejdell Sylow|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.  The textbooks of Weber and Burnside helped establish group theory as a discipline.<ref>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."</ref>  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.

==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 [[presentation of a group|presentations]] came into a life of their own through the work of Cayley, [[Walther von Dyck]], [[Max Dehn]], [[Jakob Nielsen (mathematician)|Jakob Nielsen]], [[Otto Schreier]], and continued in the 1920-1940 period with the work of [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], [[Wilhelm 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 integer|square-free]] order, and the early beginnings of the [[character theory]] of Frobenius. Already by 1860, the [[automorphism group|groups of automorphisms]] of the [[finite projective plane]]s had been studied (by Mathieu), and in the 1870s 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 [[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.

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.  <!-- When was the classification of semisimple Lie algebras finished? Here or in the next period?  Check Borel or Springer. -->

==Early 20th century==
In the period 1900–1940, infinite "discontinuous" groups (now called [[discrete group]]s) gained life of their own. [[Burnside problem|Burnside's famous problem]] ushered in the study of arbitrary subgroups of finite-dimensional [[linear group]]s 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. [[Bernard Neumann]] and [[Hanna Neumann]] produced their study of [[Variety (universal algebra)|varieties of group]]s, groups defined by group-theoretic equations rather than polynomial ones. <!-- Baer also did some frightening stuff during this time.  Maybe also mention the developments of Abelian group theory during this time. -->

Continuous groups also had explosive growth in the 1900-1940 period.  [[Topological group]]s began to be studied as such.  There were many great achievements in continuous groups: Cartan's classification of [[semisimple Lie algebra]]s, [[Hermann Weyl]]'s theory of representations of [[compact group]]s, [[Alfréd Haar]]'s work in the [[locally compact group|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 [[Philip Hall]]: on a generalization of Sylow's theorem to arbitrary sets of primes which revolutionized the study of finite [[soluble group]]s, and on the power-commutator structure of {{nowrap|[[p-group|''p''-groups]]}}, including the ideas of [[regular p-group|regular ''p''-groups]] and [[isoclinism of groups]], which revolutionized the study of {{nowrap|''p''-groups}} and was the first major result in this area since Sylow. This period saw [[Hans 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. <!-- Transfer results of Burnside, Frobenius, Schur, Grün.  Probably tons of other important things missing.  Check Wehrfritz for finite soluble. -->

==Mid-20th century==
<!-- 1940-1960 works well for several topics:
* finite groups:
** simple: includes Brauer but not Janko, missing at least some of Feit-Thompson-Suzki
** solbule: includes Wielandt, but not Carter,
* algebraic: includes Chevalley+Zariski, but I think not Tits,
* infinite:
** soluble: includes Hall, but I think not the more serious Grünberg stuff, with group rings and cohomology, check Robinson-Lennox
** check on when profinite got popular
** burnside problem: includes some progress
* continuous:
** locally compact: includes Mackey
** Lie: includes the Montgomery, Zippin regularity result

-->

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, and [[representation theory]].<ref>{{harvnb|Curtis|2003}}. (See also [[History of representation theory]]).</ref> Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to [[classification of finite simple groups|classify all finite simple groups]] in 1982. Completing and simplifying the proof of the classification are areas of active research.<ref>{{cite journal |first=Michael |last=Aschbacher |title=The Status of the Classification of the Finite Simple Groups |journal=Notices of the American Mathematical Society |volume=51 |issue=7 |pages=736–740 |year=2004 |doi= |url=https://www.ams.org/notices/200407/fea-aschbacher.pdf}}</ref>

[[Anatoly Maltsev]] also made important contributions to group theory during this time; his early work was in [[mathematical logic|logic]] in the 1930s, but in the 1940s he proved important embedding properties of [[semigroup]]s into groups, studied the isomorphism problem of [[group ring]]s, established the Malçev correspondence for [[polycyclic group]]s, and in the 1960s return to logic proving various theories within the study of groups to be [[Decidability (logic)#Decidability of a theory|undecidable]].  Earlier, [[Alfred Tarski]] proved elementary group theory undecidable.<ref>{{cite book |first=Alfred |last=Tarski |chapter=Undecidability of the elementary theory of groups |editor-first=Alfred |editor-last=Tarski |editor2-last=Mostowski |editor3-link=Raphael Robinson |editor3-first=Raphael M. |editor3-last=Robinson |title=Undecidable Theories |publisher=North-Holland |year=1953 |isbn= |pages=77–87 |series=Studies in logic and the foundations of mathematics |volume=14}}</ref>

<!-- 1960-1980
* Finite:
** Simple: all the sporadics, odd-order, full Feit, Thompson, Suzuki, Glauberman glory
** Soluble: Carter, Schunk, classes of groups
** ModCharTh: Green, Janusz, more Brauer
* Discrete:
** algebraic: Tits, Lang
** burnside: mostly finished I think
** combinatorial: Mal'cev
* Continuous:
** Profinite: definitely around, coclass conjectures made
** analytic: ?, check Mackey's history for the early part
**

-->
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 group]]s, 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 module]]s 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 [[Jacques Tits]] and the availability the surjectivity of [[Serge Lang]]'s map allowed a revolution in algebraic groups.  The [[Burnside problem]] had tremendous progress, with better [[counterexample]]s 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.

Continuous groups broadened considerably, with [[p-adic number|''p''-adic]] analytic questions becoming important.  Many [[conjecture]]s were made during this time, including the coclass conjectures.

==Late 20th century==
<!-- 1980-2000
* Finite:
** Simple/ModRepTh: fusion systems, Puig's nilpotent blocks, amalgamation, post classification results
** Soluble: Doerk-Hawkes
** p-groups: completion of coclass proof
** CGT: grows from beginnings in the 70s to a full fledged discipline
** finite geometries: some important work here in this period
* Discrete: have to check some references. lots of progress on group rings of discrete groups
** geometry: Thurston's geometrization, orbifolds in general, knot theory in general
** combinatorial: automatic groups
** geometric: hyperbolic groups
* Continuous: have to check with an expert
-->

The last twenty years of the 20th 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 subgroup]]s of the simple groups and the corresponding classifications of [[primitive group]]s.  In [[finite geometry]] and [[combinatorics]], many problems could now be settled.<!-- list a few, should be a few in that 1981 proceedings, and in Cameron's textbook, and in the Design theory encyclopedia-->  The [[modular representation theory]] entered a new era as the techniques of the classification were axiomatized, including fusion systems, Luis Puig's theory of pairs and nilpotent blocks.  The theory of finite soluble groups was likewise transformed by the influential book of Klaus Doerk and Trevor 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 [[William 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.

==Today==
<!-- 2000-today
This part should not contain any details, but just indicate that group theory's history is still being written.
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Group theory continues to be an intensely studied matter. 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.

==Notes==
{{reflist|2}}

==References==
* [[List of publications in mathematics#Group theory|Historically important publications in group theory]].
* {{Citation | last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-2677-5 | year=2003}}
* {{Citation | last=Galois | first=Évariste | author-link=Évariste Galois | editor1-last=Tannery | editor1-first=Jules | title=Manuscrits de Évariste Galois | url=http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN9280 | publisher=Gauthier-Villars | location=Paris | year=1908}}
* {{Citation | doi=10.2307/2690312 | last1=Kleiner | first1=Israel | title=The evolution of group theory: a brief survey |mr=863090 | year=1986 | journal=[[Mathematics Magazine]] | issn=0025-570X | volume=59 | issue=4 | pages=195–215 | jstor=2690312}}
* {{cite book |last1=Kleiner |first1=Israel |editor-first1=Israel |editor-last1=Kleiner |title=A history of abstract algebra |date=2007 |publisher=Birkhäuser |location=Boston, Mass. |doi=10.1007/978-0-8176-4685-1 |isbn=978-0-8176-4685-1 |url=https://link.springer.com/book/10.1007/978-0-8176-4685-1}}
* {{Citation | last1=Smith | first1=David Eugene | author1-link=David Eugene Smith | title=History of Modern Mathematics | url=https://www.gutenberg.org/ebooks/8746 | series=Mathematical Monographs, No. 1 | year=1906}}
* {{Citation | last1=Wussing | first1=Hans|authorlink=Hans Wussing| title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-45868-7 | year=2007}}
* {{Citation | last=du Sautoy | first=Marcus | author-link=Marcus du Sautoy | title=Finding Moonshine | publisher=[[Fourth Estate]] | location=London | isbn=978-0-00-721461-7 | year=2008 | url-access=registration | url=https://archive.org/details/findingmoonshine0000dusa }}

{{History of mathematics}}

[[Category:Group theory]]
[[Category:History of mathematics|Group theory]]