Group theory 41890 225741865 2008-07-15T04:49:25Z Arcfrk 3794588 /* Groups with additional structure */ rewrote {{Groups}} '''Group theory''' is a mathematical discipline, the part of [[abstract algebra]] that studies the [[algebraic structure]]s known as [[group (mathematics)|groups]]. The development of group theory sprung from three main sources: [[number theory]], theory of [[algebraic equation]]s, and [[geometry]]. The number-theoretic strand was started by [[Leonhard Euler]] and taken up by [[Carl Friedrich Gauss|Gauss]], who developed [[modular arithmetic]] and considered additive and multiplicative groups related to [[quadratic field]]s. Early results about [[permutation group]]s were obtained by [[Joseph Louis Lagrange|Lagrange]], [[Paolo Ruffini|Ruffini]], and [[Niels Henrik Abel|Abel]] in their quest for general solutions of polynomial equations of high degree. [[Évariste Galois|Galois]] coined the term “group” and established a connection between the nascent theory of groups and [[field theory (mathematics)|field theory]], which is known as [[Galois theory]]. In geometry, groups first became important in [[projective geometry]] and, later, [[non-Euclidean geometry]]. [[Felix Klein]] in his [[Erlangen program]] famously proclaimed group theory to be the organizing principle behind the very meaning of geometry. Groups manifest themselves as [[symmetry group]]s of various physical systems, such as [[crystal]]s and the [[hydrogen atom]]. Thus group theory and the closely related [[representation theory]] have many applications in [[physics]] and [[chemistry]]. The concept of a group is a central concept of abstract algebra: other algebraic structures, such as [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[vector space]]s are elaborations of groups, which are endowed with additional operations. Groups recur throughout mathematics, and methods of group theory had a strong influence on ring theory and other parts of algebra.<!--Should we mention the [[Krull–Schmidt theorem]] here?--> [[Linear algebraic group]]s and [[Lie group]]s are two classes of groups whose theory has been tremendously advanced, and became the subject areas of their own. A central question of group theory throughout much of the last century was the [[classification of finite simple groups]]. The result of a collaborative effort mostly from 1960-1980 and totaling more than ten thousand pages, it is one of the most important mathematical achievements of the 20th century. ==History== {{Main|History of group theory}} There are three historical roots of group theory: the theory of [[algebraic equation]]s, [[number theory]] and [[geometry]]. Historically, the first use of groups to determine the solvability of [[polynomial equation]]s was done by [[Évariste Galois]], in the 1830's. Investigations were pushed further, mainly in the guise of [[permutation group]]s, by [[Arthur Cayley]] and [[Augustin Louis Cauchy]]. The second historical source for groups stems from [[geometry|geometrical]] situations. In an attempt to come to grips with possible geometries (such as [[euclidean geometry|euclidean]], [[hyperbolic geometry|hyperbolic]] or [[projective geometry]]) using group theory, [[Felix Klein]] initiated the [[Erlangen programme]]. [[Sophus Lie]], in 1884, started using groups (now called [[Lie group]]s) attached to [[analysis (mathematics)|analytic]] problems. Thirdly, groups were (first implicitly and later explicitly) used in [[algebraic number theory]]. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since, the impact of group theory has been ever growing, giving rise to the birth of [[abstract algebra]] in the early 20th century, [[representation theory]], and many more influential spin-off domains. The [[classification of finite simple groups]] is a vast body of work from the mid 20th century, classifying all the [[finite set|finite]] [[simple group]]s. == Main classes of groups == {{Main|Group (mathematics)|Glossary of group theory}} The range of groups being considered has gradually expanded from finite [[permutation group]]s and special examples of [[matrix group]]s to abstract groups that may be specified through a [[presentation of a group|presentation]] by generators and relations. === Permutation groups === The first class of groups to undergo a systematic study was [[permutation group]]s. Given any set ''X'' and a collection ''G'' of bijections of ''X'' into itself (known as ''permutations'') that is closed under compositions and inverses, ''G'' is a group [[group action|acting]] on ''X''. If ''X'' consists of ''n'' elements and ''G'' consists of ''all'' permutations, ''G'' is the [[symmetric group]] ''S''<sub>''n''</sub>; in general, ''G'' is a [[subgroup]] of the symmetric group of ''X''. An early construction due to [[Arthur Cayley|Cayley]] exhibited any group as a permutation group, acting on itself (''X''&nbsp;=&nbsp;''G'') by means of the left [[regular representation]]. In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for ''n''&nbsp;&ge;&nbsp;5, the [[alternating group]] ''A''<sub>''n''</sub> is [[simple group|simple]], i.e. does not admit any proper [[normal subgroup]]s. This fact plays a key role in the [[Abel–Ruffini theorem|impossibility of solving a general algebraic equation of degree ''n''&nbsp;&ge;&nbsp;5 in radicals]]. === Matrix groups === The next important class of groups is given by ''matrix groups'', or [[linear group]]s. Here ''G'' is a set consisting of invertible [[matrix (mathematics)|matrices]] of given order ''n'' over a [[field (mathematics)|field]] ''K'' that is closed under the products and inverses. Such a group acts on the ''n''-dimensional vector space ''K''<sup>''n''</sup> by [[linear transformation]]s. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully expoited to establish properties of the group ''G''. === Transformation groups === Permutation groups and matrix groups are special cases of [[transformation group]]s: groups that act on a certain space ''X'' preserving its inherent structure. In the case of permutation groups, ''X'' is a set; for matrix groups, ''X'' is a [[vector space]]. The concept of a transformation group is closely related with the concept of a [[symmetry group]]: transformation groups frequently consist of ''all'' transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with [[differential geometry]]. A long line of research, originating with [[Sophus Lie|Lie]] and [[Felix Klein|Klein]], considers group actions on [[manifold]]s by [[homeomorphism]]s or [[diffeomorphism]]s. The groups themselves may be [[discrete group|discrete]] or [[continuous group|continuous]]. === Abstract groups === Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a [[presentation of a group|presentation]] by ''generators and relations'', : <math> G = \langle S|R\rangle. </math> A significant source of abstract groups is given by the construction of a ''factor group'', or [[quotient group]], ''G''/''H'', of a group ''G'' by a [[normal subgroup]] ''H''. [[Class group]]s of [[algebraic number field]]s were among the earliest examples of factor groups, of much interest in [[number theory]]. If a group ''G'' is a permutation group on a set ''X'', the factor group ''G''/''H'' is no longer acting on ''X''; but the idea of an abstract group permits one not to worry about this discrepancy. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under [[isomorphism]], as well as the classes of group with a given such property: [[finite group]]s, [[periodic group]]s, [[simple group]]s, [[solvable group]]s, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of [[abstract algebra]] in the works of [[David Hilbert|Hilbert]], [[Emil Artin]], [[Emmy Noether]], and mathematicians of their school. === Topological and algebraic groups === An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a [[topological space]], [[differentiable manifold]], or [[algebraic variety]]. If the group operations ''m'' (multiplication) and ''i'' (inversion), : <math> m: G\times G\to G, (g,h)\mapsto gh, \quad i:G\to G, g\mapsto g^{-1}, </math> are compatible with this structure, i.e. are [[continuous map|continuous]], [[smooth map|smooth]] or [[regular map|regular]] (in the sense of algebraic geometry) maps then ''G'' becomes a [[topological group]], a [[Lie group]], or an [[algebraic group]].<ref>This process of imposing extra structure has been formalized through the notion of a [[group object]] in a suitable [[category (mathematics)|category]]. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.</ref> The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for [[abstract harmonic analysis]], whereas [[Lie group]]s (frequently realized as transformation groups) are the mainstays of [[differential geometry]] and unitary [[representation theory]]. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, [[compact Lie group|compact connected Lie groups]] have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group ''&Gamma;'' can be realized as a [[lattice (discrete subgroup)|lattice]] in a topological group ''G'', the geometry and analysis pertaining to ''G'' yield important results about ''&Gamma;''. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups ([[profinite group]]s): for example, a single [[powerful p-group|''p''-adic analytic group]] ''G'' has a family of quotients which are finite [[p-group|''p''-groups]] of various orders, and properties of ''G'' translate into the properties of its finite quotients. ==Combinatorial and geometric group theory== Groups can be described in different ways. Finite groups can be described by writing down the [[group table]] consisting of all possible multiplications {{nowrap|''g'' • ''h''}}. A more important way of defining a group is by ''generators and relations'', also called the ''presentation'' of a group. Given any set ''F'' of generators {''g''<sub>''i''</sub>}<sub>''i'' ∈ ''I''</sub>, the [[free group]] generated by ''F'' surjects onto the group ''G''. The kernel of this map is called subgroup of relations, generated by some subset ''D''. The presentation is usually denoted by {{nowrap begin}}〈''F'' | ''D'' 〉{{nowrap end}}. For example, the group {{nowrap begin}}'''Z''' = 〈''a'' | 〉{{nowrap end}} can be generated by one element ''a'' (equal to +1 or &minus;1) and no relations, because ''n''·1 never equals 0 unless ''n'' is zero. A string consisting of generator symbols is called a ''word''. [[Combinatorial group theory]] studies groups from the perspective of generators and relations.<ref>{{harvnb|Schupp|Lyndon|2001}}</ref> It is particularly useful where finitness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of [[graph (mathematics)|graphs]] via their [[fundamental group]]s. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The ''[[word problem for groups|word problem]]'' asks whether two words are effectively the same group element. By relating the problem to [[Turing machine]]s, one can show that there is in general no [[algorithm]] solving this task. An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example '''Z''' can also be presented by :〈''x'', ''y'' | ''xyxyx'' = 1&rang; and it is not obvious (but true) that this presentation is isomorphic to the standard one above. [[Image:Cayley graph of F2.svg|right|150px|thumb|The Cayley graph of &lang; x, y ∣ &rang;, the free group of rank 2.]] [[Geometric group theory]] attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.<ref>{{harvnb|La Harpe|2000}}</ref> The first idea is made precise by means of the [[Cayley graph]], whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the [[word metric]] given by the length of the minimal path between the elements. A theorem of [[John Milnor|Milnor]] and Svarc then says that given a group ''G'' acting in a reasonable manner on a [[metric space]] ''X'', for example a [[compact manifold]], then ''G'' is [[quasi-isometry|quasi-isometric]] (i.e. looks similar from the far) to the space ''X''. ==Representation of groups== Saying that a group ''G'' ''[[group action|acts]]'' on a set ''X'' means that every element defines a bijective map on a set in a way compatible with the group structure. When ''X'' has more structure, it is useful to restrict this notion further: a representation of ''G'' on a [[vector space]] ''V'' is a group homomorphism :''&rho;'' : ''G'' &rarr; ''GL''(''V''), where ''[[general linear group|GL]]''(''V'') consists of the invertible [[linear map|linear transformations]] of ''V''. In other words, to every group element ''g'' is assigned an [[automorphism]] ''ρ''(''g'') such that {{nowrap begin}}''ρ''(''g'') ∘ ''ρ''(''h'') = ''ρ''(''gh''){{nowrap end}} for any ''h'' in ''G''. This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.<ref>Such as [[group cohomology]] or [[equivariant K-theory]].</ref> On the one hand, it may yield new information about the group ''G'': often, the group operation in ''G'' is abstractly given, but via ''ρ'', it corresponds to the [[matrix multiplication|multiplication of matrices]], which is very explicit.<ref>In particular, if the representation is [[faithful representation|faithful]].</ref> On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if ''G'' is finite, it is [[Maschke's theorem|known]] that ''V'' above decomposes into [[irreducible representation|irreducible parts]]. These parts in turn are much more easily manageable than the whole ''V'' (via [[Schur's lemma]]). Given a group ''G'', [[representation theory]] then asks what representations of ''G'' exist. There are several settings, and the employed methods and obtained results are rather different in every case: [[representation theory of finite groups]] and representations of [[Lie group]]s are two main subdomains of the theory. The totality of representations is governed by the group's [[character theory|characters]]. For example, [[Fourier series|Fourier polynomial]]s can be interpreted as the characters of [[unitary group|''U''(1)]], the group of [[complex numbers]] of [[absolute value]] ''1'', acting on the [[Lp space|''L''<sup>2</sup>]]-space of periodic functions. ==Connection of groups and symmetry== {{main|Symmetry group}} Given a structured object ''X'' of any sort, a [[symmetry]] is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example #If ''X'' is a set with no additional structure, a symmetry is a [[bijective]] map from the set to itself, giving rise to [[permutation group]]s. #If the object ''X'' is a set of points in the plane with its [[metric (mathematics)|metric]] structure or any other [[metric space]], a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an [[isometry]]). The corresponding group is called [[isometry group]] of ''X''. #If instead [[angle]]s are preserved, one speaks of [[conformal map]]s. Conformal maps give rise to [[Kleinian group]]s, for example. #Symmetries are not restricted to geometrical objects, but include algebraic objects as well: the equation ::''x''<sup>4</sup> &minus; 7''x''<sup>2</sup> + 12 = 0 :has the solutions +2, &minus;2, <math>+\sqrt{3}</math>, and <math>-\sqrt{3}</math>. Exchanging &minus;2 and +2 and the two square roots determines a group, the [[Galois group]] belonging to the equation. The axioms of a group formalize the essential aspects of [[symmetry]]. Symmetries form a group: they are [[closure (mathematics)|closed]] because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by the undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. [[Frucht's theorem]] says that every group is the symmetry group of some [[graph (mathematics)|graph]]. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a [[category (mathematics)|category]]. Maps preserving the structure are then the [[morphisms]], and the symmetry group is the [[automorphism group]] of the object in question. ==Applications of group theory==<!--this section is linked at from [[Applications of group theory]]--> Applications of group theory abound. Almost all structures in [[abstract algebra]] are special cases of groups. [[Ring (mathematics)|Rings]], for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore group theoretic arguments underlie large parts of the theory of those entities. [[Galois theory]] uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The [[fundamental theorem of Galois theory]] provides a link between [[algebraic field extension]]s and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding [[Galois group]]. For example, ''S''<sub>5</sub>, the [[symmetric group]] in 5 elements, is not solvable which implies that the general [[quintic equation]] cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as [[class field theory]]. [[Algebraic topology]] is another domain which prominently [[functor|associates]] groups to the objects the theory is interested in. There, groups are used to describe certain invariants of [[topological space]]s. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some [[homeomorphism|deformation]]. For example, the [[fundamental group]] "counts" how many paths in the space are essentially different. The [[Poincaré conjecture]], proved in 2002/2003 by [[Perelman]] is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of [[Eilenberg-MacLane space]]s which are spaces with prescribed [[homotopy groups]]. Similarly [[algebraic K-theory]] stakes in a crucial way on [[classifying space]]s of groups. Finally, the name of the [[torsion subgroup]] of an infinite group shows the legacy of topology in group theory. [[Image:Torus.png|thumb|right|200px|A torus. Its abelian group structure is induced from the map {{nowrap begin}}'''C''' &rarr; '''C'''/'''Z'''+''&tau;'''''Z'''{{nowrap end}}, where ''&tau;'' is a parameter.]] [[Image:Caesar3.svg|thumb|left|150px|The [[additive group]] '''Z'''/26 underlies Caesar's cipher.]] [[Algebraic geometry]] and [[cryptography]] likewise uses group theory in many ways. [[Abelian varieties]] have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.<ref>For example the [[Hodge conjecture]] (in certain cases).</ref> The one-dimensional case, namely [[elliptic curve]]s is studied in particular detail. They are both theoretically and practically intriguing.<ref>See the [[Birch-Swinnerton-Dyer conjecture]], one of the [[millennium problem]]s</ref> Very large groups of prime order constructed in [[Elliptic Curve Cryptography|Elliptic-Curve Cryptography]] serve for [[public key cryptography]]. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the [[discrete logarithm]] very hard to calculate. One of the earliest encryption protocols, [[Caesar's cipher]], may also be interpreted of a (very easy) group operation. In another direction, [[Toric variety|toric varieties]] are [[algebraic variety|algebraic varieties]] acted on by a [[torus]]. Toroidal embeddings have recently led to advances in [[algebraic geometry]], in particular [[resolution of singularities]].<ref>{{Citation | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Wlodarczyk | first4=Jaroslaw | title=Torification and factorization of birational maps | id={{MathSciNet | id = 1896232}} | year=2002 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=15 | issue=3 | pages=531–572}}</ref> [[Algebraic number theory]] would not exist without group theory. For example, [[Euler product|Euler's product formula]] :<math> \begin{align} \sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \\ \end{align} \!</math> captures [[Fundamental theorem of arithmetic|the fact]] that any integer decomposes in a unique way into [[prime number|primes]]. The failure of this statement for [[Dedekind ring|more general rings]] gives rise to [[class group]]s and [[regular prime]]s, which feature in [[Ernst Kummer|Kummer's]] treatment of [[Fermat's last theorem]]. *The concept of the [[Lie group]] (named after mathematician [[Sophus Lie]]) is important in the study of [[differential equations]] and [[manifold]]s; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called [[harmonic analysis]]. [[Haar measure]]s, that is integrals invariant under the translation in a Lie group, are used for [[pattern recognition]] and other [[image processing]] techniques.<ref>{{Citation | last1=Lenz | first1=Reiner | title=Group theoretical methods in image processing | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-0-387-52290-6 | year=1990 | volume=413}}</ref> *In [[combinatorics]], the notion of [[permutation]] group and the concept of group action are often used to simplify the counting of a set of objects; see in particular [[Burnside's lemma]]. [[Image:Fifths.png|right|thumb|150px|The circle of fifths may be endowed with a cyclic group structure]] *The presence of the 12-[[periodicity]] in the [[circle of fifths]] yields applications of [[elementary group theory]] in [[musical set theory]]. *An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of [[Lie group]]s, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include: [[Standard Model]], [[Gauge theory]], [[Lorentz group]], [[Poincaré group]] *In [[chemistry]], groups are used to classify crystal structures, regular polyhedra, and the [[molecular symmetry|symmetries of molecules]]. The assigned point groups can then be used to determine physical properties (such as [[Polarity (physics)|polarity]] and [[Chirality (chemistry)|chirality]]), spectroscopic properties (particularly useful for [[Raman spectroscopy]] and [[Infrared spectroscopy]]), and to construct molecular orbitals. == See also == *[[group (mathematics)]] *[[Glossary of group theory]] *[[List of group theory topics]] ==Notes== <references/> ==References== * {{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | title=Linear algebraic groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-97370-8 | id={{MathSciNet | id = 1102012}} | year=1991 | volume=126}} * {{Citation | last1=Cannon | first1=John J. | title=Computers in group theory: A survey | id={{MathSciNet | id = 0290613}} | year=1969 | journal=Communications of the Association for Computing Machinery | issn=0001-0782 | volume=12 | pages=3–12}} *Connell, Edwin, ''[http://www.math.miami.edu/~ec/book/ Elements of Abstract and Linear Algebra.]'' Free online textbook. * {{Citation | last1=Frucht | first1=R. | title=Herstellung von Graphen mit vorgegebener abstrakter Gruppe | url=http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0 | year=1939 | journal=Compositio Mathematica | issn=0010-437X | volume=6 | pages=239–50}} * {{Citation | last1=Kleiner | first1=Israel | title=The evolution of group theory: a brief survey | id={{MathSciNet | id = 863090}} | year=1986 | journal=[[Mathematics Magazine]] | issn=0025-570X | volume=59 | issue=4 | pages=195–215}} * {{Citation | last1=La Harpe | first1=Pierre de | title=Topics in geometric group theory | publisher=[[University of Chicago Press]] | isbn=978-0-226-31721-2 | year=2000}} *{{cite book | author=Livio, M. | title= The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry | publisher=Simon & Schuster | year=2005 | id=ISBN 0-7432-5820-7}} A pleasant read, explaining the importance of group theory and how its symmetries point to symmetries in physics and other sciences. Conveys well the practical value of group theory. * {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | publisher=[[Oxford University Press]] | isbn=978-0-19-560528-0 | oclc=138290 | year=1970}} *{{cite book | author=Rotman, Joseph | title=An introduction to the theory of groups | location=New York | publisher=Springer-Verlag | year=1994 | id=ISBN 0-387-94285-8}} A standard contemporary reference. * {{Citation | last1=Weibel | first1=Charles A. | title=An introduction to homological algebra | publisher=[[Cambridge University Press]] | isbn=978-0-521-55987-4 | id={{MathSciNet | id = 1269324}} | year=1994}} * {{Citation | last1=Schupp | first1=Paul E. | last2=Lyndon | first2=Roger C. | title=Combinatorial group theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-41158-1 | year=2001}} *{{cite book | author=Scott, W. R. | title= Group Theory | location=New York | publisher=Dover | year=1987 | origyear=1964 | id=ISBN 0-486-65377-3}} Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. * {{Citation | last1=Shatz | first1=Stephen S. | title=Profinite groups, arithmetic, and geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08017-8 | id={{MathSciNet | id = 0347778}} | year=1972}} ==External links== * [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html History of the abstract group concept] * [http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory] This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems. 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