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The manipulations of this form the. In , a group is a equipped with a that combines any two to form a third element in such a way that four conditions called group are satisfied, namely , , and. One of the most familiar examples of a group is the set of together with the operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of. For example, a encodes symmetry features of a object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. are the symmetry groups used in the of ; , which are also Lie groups, can express the physical symmetry underlying ; and are used to help understand. The concept of a group arose from the study of , starting with in the 1830s, who introduced the term of group groupe, in French for the symmetry group of the of an equation, now called a. After contributions from other fields such as and geometry, the group notion was generalized and firmly established around 1870. Modern —an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as , and. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of that is, through the and of. A theory has been developed for , which culminated with the , completed in 2004. Since the mid-1980s, , which studies as geometric objects, has become an active area in group theory. , together with. The following properties of integer addition serve as a model for the group axioms given in the definition below. That is, addition of integers always yields an integer. This property is known as under addition. Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as. is called the of addition because adding it to any integer returns the same integer. To appropriately understand these structures as a collective, the following is developed. Definition [ ] The axioms for a group are short and natural... Yet somehow hidden behind these axioms is the , a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. Such an element is unique , and thus one speaks of the identity element. The result of the group operation may depend on the order of the operands. The symmetry group described in the following section is an example of a group that is not abelian. The identity element of a group G is often written as 1 or 1 G, a notation inherited from the. The identity element can also be written as id. Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set. As this formulation of the definition avoids , it is generally preferred for and for. This formulation exhibits groups as a variety of. It is also useful for talking of properties of the inverse operation, as needed for defining and. Second example: a symmetry group [ ] Two figures in the plane are if one can be changed into the other using a combination of , , and. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called. A square has eight symmetries. These are: The elements of the symmetry group of the square D 4. Vertices are identified by color or number. the leaving everything unchanged, denoted id;• reflections about the horizontal and vertical middle line f v and f h , or through the two f d and f c. These symmetries are. Each sends a point in the square to the corresponding point under the symmetry. two of these symmetries gives another symmetry. These symmetries determine a group called the of degree 4, denoted D 4. The underlying set of the group is the above set of symmetries, and the group operation is. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. This is the usual notation for composition of functions. The on the right lists the results of all such compositions possible. of D 4 id r 1 r 2 r 3 f v f h f d f c id id r 1 r 2 r 3 f v f h f d f c r 1 r 1 r 2 r 3 id f c f d f v f h r 2 r 2 r 3 id r 1 f h f v f c f d r 3 r 3 id r 1 r 2 f d f c f h f v f v f v f d f h f c id r 2 r 1 r 3 f h f h f c f v f d r 2 id r 3 r 1 f d f d f h f c f v r 3 r 1 id r 2 f c f c f v f d f h r 1 r 3 r 2 id The elements id, r 1, r 2, and r 3 form a , highlighted in red upper left region. A left and right of this subgroup is highlighted in green in the last row and yellow last column , respectively. Given this set of symmetries and the described operation, the group axioms can be understood as follows:• Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table. The associativity constraint deals with composing more than two symmetries: Starting with three elements a, b and c of D 4, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose a and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with a. , a product of many group elements can be simplified in any grouping. While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. An inverse element undoes the transformation of some other element. In other words, D 4 is not abelian, which makes the group structure more difficult than the integers introduced first. History [ ] Main article: The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of of degree higher than 4. The 19th-century French mathematician , extending prior work of and , gave a criterion for the solvability of a particular polynomial equation in terms of the of its solutions. The elements of such a correspond to certain of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general were investigated in particular by. Geometry was a second field in which groups were used systematically, especially symmetry groups as part of 's 1872. After novel geometries such as and had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, founded the study of in 1884. The third field contributing to group theory was. Certain structures had been used implicitly in ' number-theoretical work 1798 , and more explicitly by. In 1847, made early attempts to prove by developing into. 1882 introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of and , who worked on of finite groups, 's and 's papers. The theory of Lie groups, and more generally was studied by , and many others. Its algebraic counterpart, the theory of , was first shaped by from the late 1930s and later by the work of and. The 's 1960—61 Group Theory Year brought together group theorists such as , and , laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the , with the final step taken by and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch, impacting many other fields. Elementary consequences of the group axioms [ ] Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. The axioms may be weakened to assert only the existence of a and. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above. Uniqueness of identity element and inverses [ ] Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element. In other words, there is only one inverse element of a. Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements e and f. In this case, the group operation is often denoted as an , and one talks of subtraction and difference instead of division and quotient. A consequence of this is that multiplication by a group element g is a. This function is called the left translation by g. If G is abelian, the left and the right translation by a group element are the same. To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups which sets, being "structureless", do not have , constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a , in this case the. Group homomorphisms [ ] Main article: Group homomorphisms are functions that preserve group structure. In other words, the result is the same when performing the group operation after or before applying the map a. Thus a group homomorphism respects all the structure of G provided by the group axioms. From an abstract point of view, isomorphic groups carry the same information. Subgroups [ ] Main article: Informally, a subgroup is a group H contained within a bigger one, G. Knowing is important in understanding the group as a whole. Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S. Again, this is a subgroup, because combining any two of these four elements or their inverses which are, in this particular case, these same elements yields an element of this subgroup. Cosets [ ] Main article: In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D 4 above, once a reflection is performed, the square never gets back to the r 2 configuration by just applying the rotation operations and no further reflections , i. , the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. The left cosets of any subgroup H form a of G; that is, the of all left cosets is equal to G and two left cosets are either equal or have an. , if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i. Quotient groups [ ] Main article: In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. For this to be possible, the subgroup has to be. The group operation on the quotient is shown at the right. Quotient groups and subgroups together form a way of describing every group by its : any group is the quotient of the over the of the group, quotiented by the subgroup of relations. A presentation of a group can also be used to construct the , a device used to graphically capture. , any element of the target has at most one. Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. and of group homomorphisms and the address this phenomenon. Examples and applications [ ] The fundamental group of a plane minus a point bold consists of loops around the missing point. This group is isomorphic to the integers. Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition is considered, one obtains. These groups are predecessors of important constructions in. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by groups to them and studying the properties of the corresponding groups. For example, founded what is now called by introducing the. By means of this connection, such as and translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered and thus irrelevant , because it can be to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop or any other loop around the hole. This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, employs geometric concepts, for example in the study of. Further branches crucially applying groups include and. In addition to the above theoretical applications, many practical applications of groups exist. relies on the combination of the abstract group theory approach together with knowledge obtained in , in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as , and benefit from the concept. Numbers [ ] Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as and. Further concepts such as , and also form groups. Rationals [ ] The desire for the existence of multiplicative inverses suggests considering a b. Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. The rational numbers including 0 also form a group under addition. Group theoretic arguments therefore underlie parts of the theory of those entities. Modular arithmetic [ ] The hours on a clock form a group that uses 12. In , two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the of that division. This is familiar from the addition of hours on the face of a : if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. For any p, there is also the. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication. The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. The inverse b can be found by using and the fact that the gcd a, p equals 1. Hence all group axioms are fulfilled. They are crucial to. Cyclic groups [ ] The 6th complex roots of unity form a cyclic group. z is a primitive element, but z 2 is not, because the odd powers of z are not a power of z 2. A cyclic group is a group all of whose elements are of a particular element a. In multiplicative notation, the elements of the group are:... Such an element a is called a generator or a of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as... Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. The group operation is multiplication of complex numbers. Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element a, all the powers of a are distinct; despite the name "cyclic group", the powers of the elements do not cycle. As these two prototypes are both abelian, so is any cyclic group. The study of finitely generated abelian groups is quite mature, including the ; and reflecting this state of affairs, many group-related notions, such as and , describe the extent to which a given group is not abelian. Symmetry groups [ ] See also: , , and Symmetry groups are groups consisting of of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as and their solutions. Conceptually, group theory can be thought of as the study of symmetry. greatly simplify the study of or. A group is said to on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the acts on the tiling by permuting the highlighted warped triangles and the other ones, too. By a group action, the group pattern is connected to the structure of the object being acted on. Rotations and reflections form the symmetry group of a. In chemical fields, such as , and describe and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule. Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a , for example, from a cubic to a tetrahedral crystalline form. An example is materials, where the change from a paraelectric to a ferroelectric state occurs at the and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft mode, a vibrational lattice mode that goes to zero frequency at the transition. Such has found further application in elementary particle physics, where its occurrence is related to the appearance of. displays , though the double bonds reduce this to. features. The 2,3,7 triangle group, a hyperbolic group, acts on this of the plane. Finite symmetry groups such as the are used in , which is in turn applied in of transmitted data, and in. Another application is , which characterizes functions having of a prescribed form, giving group-theoretic criteria for when solutions of certain are well-behaved. Geometric properties that remain stable under group actions are investigated in. General linear group and representation theory [ ] Two the left illustration multiplied by matrices the middle and right illustrations. consist of together with. The general linear group GL n, R consists of all n-by- n matrices with entries. Its subgroups are referred to as matrix groups or. The dihedral group example mentioned above can be viewed as a very small matrix group. Another important matrix group is the SO n. It describes all possible rotations in n dimensions. Via , are used in. Representation theory is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its on other spaces. A broad class of are linear representations, i. , the group is acting on a , such as the three-dimensional R 3. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. Given a group action, this gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, and , especially locally. Galois groups [ ] Main article: Galois groups were developed to help solve by capturing their symmetry features. , permuting the two solutions of the equation can be viewed as a very simple group operation. Similar formulae are known for and , but do not exist in general for and higher. Abstract properties of Galois groups associated with polynomials in particular their give a criterion for polynomials that have all their solutions expressible by radicals, i. , solutions expressible using solely addition, multiplication, and similar to the formula above. The problem can be dealt with by shifting to and considering the of a polynomial. Modern generalizes the above type of Galois groups to and establishes—via the —a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics. Finite groups [ ] Main article: A group is called finite if it has a. The number of elements is called the of the group. An important class is the S N, the groups of of N letters. For example, the symmetric group on 3 letters is the group consisting of all possible orderings of the three letters ABC, i. , contains the elements ABC, ACB, BAC, BCA, CAB, CBA, in total 6 of 3 elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S N for a suitable integer N, according to. Parallel to the group of symmetries of the square above, S 3 can also be interpreted as the group of symmetries of an. In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: states that for a finite group G the order of any finite subgroup H the order of G. The give a partial converse. The discussed above is a finite group of order 8. The order of r 1 is 4, as is the order of the subgroup R it generates see above. The order of the reflection elements f v etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. Classification of finite simple groups [ ] Main article: Mathematicians often strive for a complete or list of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic abelian groups Z p. can be used to , but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called if its only normal subgroups are the and the group itself. The exhibits finite simple groups as the building blocks for all finite groups. was a major achievement in contemporary group theory. 1998 winner succeeded in proving the conjectures, a surprising and deep relation between the largest finite simple —the ""—and certain , a piece of classical , and , a theory supposed to unify the description of many physical phenomena. Groups with additional structure [ ] Many groups are simultaneously groups and examples of other mathematical structures. In the language of , they are in a , meaning that they are objects that is, examples of another mathematical structure which come with transformations called that mimic the group axioms. For example, every group as defined above is also a set, so a group is a group object in the. Topological groups [ ] The in the under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every , such as the red arc in the figure, looks like a part of the shown at the bottom. Main article: Some may be endowed with a group law. Such groups are called topological groups, and they are the group objects in the. All of these groups are , so they have and can be studied via. The former offer an abstract formalism of invariant. Matrix groups over these fields fall under this regime, as do and , which are basic to number theory. Galois groups of infinite field extensions such as the can also be equipped with a topology, the so-called , which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of , is the. Lie groups [ ] Main article: Lie groups in honor of are groups which also have a structure, i. , they are spaces some of the appropriate. Again, the additional structure, here the manifold structure, has to be compatible, i. , the maps corresponding to multiplication and the inverse have to be. Lie groups are of fundamental importance in modern physics: links continuous symmetries to. , as well as in and are basic symmetries of the laws of. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example are the , which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of. The latter serves—in the absence of significant —as a model of in. The full symmetry group of Minkowski space, i. , including translations, is known as the. By the above, it plays a pivotal role in special relativity and, by implication, for. are central to the modern description of physical interactions with the help of. Generalizations [ ] Group-like structures Unneeded Required Unneeded Unneeded Unneeded Unneeded Required Required Unneeded Unneeded Unneeded Required Required Required Unneeded Required Unneeded Unneeded Unneeded Unneeded Required Unneeded Unneeded Required Unneeded Required Unneeded Required Unneeded Unneeded Required Unneeded Required Required Unneeded Required Required Unneeded Unneeded Unneeded Required Required Unneeded Required Unneeded Required Required Required Unneeded Unneeded Required Required Required Unneeded Required Required Required Required Required Unneeded Required Required Required Required Required , which is used in many sources, is an equivalent axiom to totality, though defined differently. In , more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a. They arise in the study of more complicated forms of symmetry, often in and structures, such as the or. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary one i. , an operation taking n arguments. With the proper generalization of the group axioms this gives rise to an. The table gives a list of several structures generalizing groups. a: lists 3,224 research papers on group theory and its generalizations written in 2005. aa: The classification was announced in 1983, but gaps were found in the proof. See for further information. Some authors therefore omit this axiom. However, group constructions often start with an operation defined on a superset, so a closure step is common in proofs that a system is a group. Lang c: See, for example, the books of Lang 2002, 2005 and Herstein 1996, 1975. d: However, a group is not determined by its lattice of subgroups. See Suzuki. e: The fact that the group operation extends this is an instance of a. f: For example, if G is finite, then the of any subgroup and any quotient group divides the size of G, according to Lagrange's theorem. i: See the for an example. j: An example is of a group which equals the of its. 1, p. l: The transition from the integers to the rationals by adding fractions is generalized by the. m: The same is true for any F instead of Q. 1, p. n: For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang , Theorem IV. The notions of of a and are other instances of this principle. o: The stated property is a possible definition of prime numbers. See. p: For example, the protocol uses the. q: The groups of order at most 2000 are known. isomorphism, there are about 49 billion. r: The gap between the classification of simple groups and the one of all groups lies in the , a problem too hard to be solved in general. See Aschbacher , p. 737. s: Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See Michler , Carter. t: More rigorously, every group is the symmetry group of some ; see , Frucht. u: More precisely, the action on the of solutions of the differential equations is considered. See Kuga , pp. 105—113. v: See for an example where symmetry greatly reduces the complexity of physical systems. w: This was crucial to the classification of finite simple groups, for example. See Aschbacher. x: See, for example, for the impact of a group action on. A more involved example is the action of an on. y: Injective and surjective maps correspond to and , respectively. They are interchanged when passing to the. Citations [ ]• 1, p. 1: "The idea of a group is one which pervades the whole of mathematics both pure and applied. Lang , App. 2, p. 360• Cook, Mariana R. 2009 , , Princeton, N. : Princeton University Press, p. 24,• 1, p. 6, p. Wussing• Kleiner• Smith• Galois• Kleiner , p. 202• Cayley• Lie• Kleiner , p. 204• Jordan• von Dyck• Curtis• Mackey• Borel• Aschbacher• 2, pp. 4—5• 1, p. 2, p. 1, p. 3, p. 1, p. 12, p. 4, p. 2, p. 4, p. 2, p. Hatcher , Chapter I, p. 12 and I. Seress• Lang , Chapter VII• Rosen , p. 54 Theorem 2. 1, p. 292• 1, p. 2, p. 1, p. 22 example 11• 5, p. 26, 29• Weyl• See also Bishop• Bersuker, Isaac 2006 , , Cambridge University Press, p. Dove, Martin T 2003 , Structure and Dynamics: an atomic view of materials, Oxford University Press, p. 265,• Welsh• Lay• Kuipers• Serre• Rudin• Robinson , p. viii• Artin• Lang , Chapter VI see in particular p. 273 for concrete examples• Lang , p. 292 Theorem VI. Artin , Theorem 6. See also Lang , p. 77 for similar results. 3, p. Ronan• Husain• Neukirch• Shatz• Milne• Warner• Borel• Goldstein• Weinberg• Naber• Becchi• Dudek References [ ] General references [ ]• 1991 , Algebra, , , Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. 2000 , The Language of Mathematics: Making the Invisible Visible, Owl Books, , Chapter 5 provides a layman-accessible explanation of groups. 1967 , Applied group theory, American Elsevier Publishing Co. , Inc. , New York, , an elementary introduction. 1996 , Abstract algebra 3rd ed. , Upper Saddle River, NJ: Prentice Hall Inc. , ,. Herstein, Israel Nathan 1975 , Topics in algebra 2nd ed. , Lexington, Mass. : Xerox College Publishing,. 2002 , Algebra, , 211 Revised third ed. , New York: Springer-Verlag, ,• Lang, Serge 2005 , Undergraduate Algebra 3rd ed. , Berlin, New York: ,. Ledermann, Walter 1953 , Introduction to the theory of finite groups, Oliver and Boyd, Edinburgh and London,. Ledermann, Walter 1973 , Introduction to group theory, New York: Barnes and Noble,. Robinson, Derek John Scott 1996 , A course in the theory of groups, Berlin, New York: Springer-Verlag,. Special references [ ]• 1998 , Galois Theory, New York: ,. 2004 , PDF , , 51 7 : 736—740. Becchi, C. 1997 , Introduction to Gauge Theories, p. 5211, : , :. Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. 2001 , , Electronic Research Announcements of the American Mathematical Society, 7: 1—4, : ,. Bishop, David H. 1993 , Group theory and chemistry, New York: Dover Publications,. 1991 , Linear algebraic groups, Graduate Texts in Mathematics, 126 2nd ed. , Berlin, New York: , ,. 1989 , Simple groups of Lie type, New York: ,. ; Delgado Friedrichs, Olaf; Huson, Daniel H. Coornaert, M. ; Delzant, T. ; Papadopoulos, A. Denecke, Klaus; Wismath, Shelly L. 2002 , Universal algebra and applications in theoretical computer science, London: ,. Dudek, W. 2001 , "On some old problems in n-ary groups", Quasigroups and Related Systems, 8: 15—36. 1939 , , Compositio Mathematica in German , 6: 239—50, archived from on 2008-12-01. ; 1991 , Representation theory. A first course, , Readings in Mathematics, 129, New York: Springer-Verlag, ,• 1980 , 2nd ed. , Reading, MA: Addison-Wesley Publishing, pp. 588—596,. 2002 , , ,. Husain, Taqdir 1966 , Introduction to Topological Groups, Philadelphia: W. Saunders Company,• ; 1937 , "Stability of Polyatomic Molecules in Degenerate Electronic States. Orbital Degeneracy", , 161 905 : 220—235, :, :. Kuipers, Jack B. 1999 , Quaternions and rotation sequences—A primer with applications to orbits, aerospace, and virtual reality, , ,. Kurzweil, Hans; Stellmacher, Bernd 2004 , The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, ,. Lay, David 2003 , Linear Algebra and Its Applications, ,. 1998 , 2nd ed. , Berlin, New York: Springer-Verlag,. Michler, Gerhard 2006 , Theory of finite simple groups, Cambridge University Press,. Milne, James S. 1980 , , Princeton University Press,• ; Fogarty, J. ; Kirwan, F. 1994 , Geometric invariant theory, 34 3rd ed. , Berlin, New York: Springer-Verlag, ,. Naber, Gregory L. 2003 , The geometry of Minkowski spacetime, New York: Dover Publications, ,. 1999 , Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, , ,• Romanowska, A. ; Smith, J. 2002 , Modes, ,. 2007 , Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, ,. Rosen, Kenneth H. 2000 , Elementary number theory and its applications 4th ed. , Addison-Wesley, ,. 1990 , Fourier Analysis on Groups, Wiley Classics, Wiley-Blackwell,. 1977 , , Berlin, New York: Springer-Verlag, ,. Shatz, Stephen S. 1972 , Profinite groups, arithmetic, and geometry, Princeton University Press, ,• 1951 , "On the lattice of subgroups of finite groups", , 70 2 : 345—371, : ,. Warner, Frank 1983 , Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag,. Welsh, Dominic 1989 , Codes and cryptography, Oxford: Clarendon Press,. 1952 , Symmetry, Princeton University Press,. Historical references [ ] See also:• 2001 , Essays in the History of Lie Groups and Algebraic Groups, Providence, R. : ,• 1889 , , II 1851—1860 ,. ; , , ,. 2003 , Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R. : American Mathematical Society,. 1882 , , in German , 20 1 : 1—44, :, archived from on 2014-02-22. 1908 , Tannery, Jules ed. , in French , Paris: Gauthier-Villars Galois work was first published by in 1843. 1870 , in French , Paris: Gauthier-Villars. 1986 , "The Evolution of Group Theory: A Brief Survey", , 59 4 : 195—215, :, ,. 1973 , Gesammelte Abhandlungen. Band 1 [Collected papers. Volume 1] in German , New York: Johnson Reprint Corp. 1976 , The theory of unitary group representations, ,• 1906 , , Mathematical Monographs, No. 2007 , The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: ,. External links [ ]• at the.
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