
Group (mathematics)

In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while being associative as well as having an identity element and inverse elements. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.
Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical 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. Lie groups arise as symmetry groups in geometry but appear also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term of "group" (, in French) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—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 subgroups, quotient groups and simple groups. 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 representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.
Definition and illustration.
First example: the integers.
One of the more familiar groups is the set of integers 
formula_1
together with addition. For any two integers formula_2 and formula_3, the sum formula_4 is also an integer; this "closure" property says that formula_5 is a binary operation on formula_6. The following properties of integer addition serve as a model for the group axioms in the definition below.
The integers, together with the operation formula_5, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.
Definition.
A group is a set formula_28 together with a binary operation on formula_28, here denoted "formula_30", that combines any two elements formula_2 and formula_3 to form an element of formula_28, denoted formula_34, such that the following three requirements, known as "group axioms", are satisfied:
Notation and terminology.
Formally, the group is the ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the "underlying set" of the group, and the operation is called the "group operation" or the "group law".
A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.
For example, consider the set of real numbers formula_57, which has the operations of addition formula_4 and multiplication formula_59. Formally, formula_57 is a set, formula_61 is a group, and formula_62 is a field. But it is common to write formula_57 to denote any of these three objects.
The "additive group" of the field formula_57 is the group whose underlying set is formula_57 and whose operation is addition. The "multiplicative group" of the field formula_57 is the group formula_67 whose underlying set is the set of nonzero real numbers formula_68 and whose operation is multiplication.
More generally, one speaks of an "additive group" whenever the group operation is notated as addition; in this case, the identity is typically denoted formula_69, and the inverse of an element formula_70 is denoted formula_71. Similarly, one speaks of a "multiplicative group" whenever the group operation is notated as multiplication; in this case, the identity is typically denoted formula_72, and the inverse of an element formula_70 is denoted formula_74. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, formula_59 instead of formula_34.
The definition of a group does not require that formula_77 for all elements formula_2 and formula_3 in formula_28. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.
Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often function composition formula_81; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol formula_82 is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
Second example: a symmetry group.
Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:
These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, formula_83 sends a point to its rotation 90° clockwise around the square's center, and formula_87 sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the dihedral group of degree four, denoted formula_92. The underlying set of the group is the above set of symmetries, and the group operation is function composition. 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. The result of performing first formula_2 and then formula_3 is written symbolically "from right to left" as formula_95 ("apply the symmetry formula_3 after performing the symmetry formula_2"). This is the usual notation for composition of functions.
The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise (formula_85) and then reflecting horizontally (formula_87) is the same as performing a reflection along the diagonal (formula_88). Using the above symbols, highlighted in blue in the group table:
formula_101
Given this set of symmetries and the described operation, the group axioms can be understood as follows.
"Binary operation": Composition is a binary operation. That is, formula_102 is a symmetry for any two symmetries formula_2 and formula_3. For example,
formula_105
that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (formula_89). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
"Associativity": The associativity axiom deals with composing more than two symmetries: Starting with three elements formula_2, formula_3 and formula_9 of formula_92, 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 formula_2 and formula_3 into a single symmetry, then to compose that symmetry with formula_9. The other way is to first compose formula_3 and formula_9, then to compose the resulting symmetry with formula_2. These two ways must give always the same result, that is,
formula_117
For example, formula_118 can be checked using the group table:
formula_119
"Identity element": The identity element is formula_120, as it does not change any symmetry formula_2 when composed with it either on the left or on the right.
"Inverse element": Each symmetry has an inverse: formula_120, the reflections formula_87, formula_86, formula_88, formula_89 and the 180° rotation formula_84 are their own inverse, because performing them twice brings the square back to its original orientation. The rotations formula_85 and formula_83 are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.
In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in formula_92, as, for example, formula_131 but formula_132. In other words, formula_92 is not abelian.
History.
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 polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's "On the theory of groups, as depending on the symbolic equation formula_134" (1854) gives the first abstract definition of a finite group.
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.
The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work "Disquisitiones Arithmeticae" (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers.
The convergence of these various sources into a uniform theory of groups started with Camille Jordan's (1870). Walther von Dyck (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 Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.
The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. These days, group theory is still a highly active mathematical branch, impacting many other fields, as the examples below illustrate.
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". For example, repeated applications of the associativity axiom show that the unambiguity of
formula_135
generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.
Individual axioms may be "weakened" to assert only the existence of a left identity and left inverses. From these "one-sided axioms", one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.
Uniqueness of identity element.
The group axioms imply that the identity element is unique: If formula_40 and formula_137 are identity elements of a group, then formula_138. Therefore it is customary to speak of "the" identity.
Uniqueness of inverses.
The group axioms also imply that the inverse of each element is unique: If a group element formula_2 has both formula_3 and formula_9 as inverses, then
Therefore it is customary to speak of "the" inverse of an element.
Division.
Given elements formula_2 and formula_3 of a group formula_28, there is a unique solution formula_70 in formula_28 to the equation formula_147, namely formula_148. (One usually avoids using fraction notation formula_149 unless formula_28 is abelian, because of the ambiguity of whether it means formula_148 or formula_152.) It follows that for each formula_2 in formula_28, the function formula_155 that maps each formula_70 to formula_157 is a bijection; it is called "left multiplication by formula_2" or "left translation by formula_2".
Similarly, given formula_2 and formula_3, the unique solution to formula_162 is formula_152. For each formula_2, the function formula_155 that maps each formula_70 to formula_167 is a bijection called "right multiplication by formula_2" or "right translation by formula_2".
Basic concepts.
When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation.
When studying groups, one instead subgroups, homomorphisms, and quotient groups. These are the appropriate analogues that take into account the existence of the group structure.
Group homomorphisms.
Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A "homomorphism" from a group formula_170 to a group formula_171 is a function formula_172 such that
It would be natural to require also that formula_173 respect identities, formula_174, and inverses, formula_175 for all formula_2 in formula_28. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.
The "identity homomorphism" of a group formula_28 is the homomorphism formula_179 that maps each element of formula_28 to itself.
An "inverse homomorphism" of a homomorphism formula_172 is a homomorphism formula_182 such that formula_183 and formula_184, that is, such that formula_185 for all formula_186 in formula_28 and such that formula_188 for all formula_189 in formula_190.
An "isomorphism" is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism.
Groups formula_28 and formula_190 are called "isomorphic" if there exists an isomorphism formula_172. In this case, formula_190 can be obtained from formula_28 simply by renaming its elements according to the function formula_173; then any statement true for formula_28 is true for formula_190, provided that any specific elements mentioned in the statement are also renamed.
The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.
Subgroups.
Informally, a "subgroup" is a group formula_190 contained within a bigger one, formula_28: it has a subset of the elements of formula_28, with the same operation. Concretely, this means that the identity element of formula_28 must be contained in formula_190, and whenever formula_204 and formula_205 are both in formula_190, then so are formula_207 and formula_208, so the elements of formula_190, equipped with the group operation on formula_28 restricted to formula_190, indeed form a group.
In the example of symmetries of a square, the identity and the rotations constitute a subgroup formula_212, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset "H" of a group "G" to be a subgroup: it is sufficient to check that formula_213 for all elements formula_186 and formula_189 in formula_190. Knowing a group's subgroups is important in understanding the group as a whole.
Given any subset formula_217 of a group formula_28, the subgroup generated by formula_217 consists of products of elements of formula_217 and their inverses. It is the smallest subgroup of formula_28 containing formula_217. In the example of symmetries of a square, the subgroup generated by formula_84 and formula_86 consists of these two elements, the identity element formula_120, and the element formula_226. 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.
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 the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup formula_190 determines left and right cosets, which can be thought of as translations of formula_190 by an arbitrary group element formula_186. In symbolic terms, the "left" and "right" cosets of formula_190, containing an element formula_186, are
The left cosets of any subgroup formula_190 form a partition of formula_28; that is, the union of all left cosets is equal to formula_28 and two left cosets are either equal or have an empty intersection. The first case formula_235 happens precisely when formula_236, i.e., when the two elements differ by an element of formula_190. Similar considerations apply to the right cosets of formula_190. The left cosets of formula_190 may or may not be the same as its right cosets. If they are (that is, if all formula_186 in formula_28 satisfy formula_242), then formula_190 is said to be a "normal subgroup".
In formula_92, the group of symmetries of a square, with its subgroup formula_245 of rotations, the left cosets formula_246 are either equal to formula_245, if formula_186 is an element of formula_245 itself, or otherwise equal to formula_250 (highlighted in green in the group table of formula_92). The subgroup formula_245 is normal, because formula_253 and similarly for the other elements of the group. (In fact, in the case of formula_92, the cosets generated by reflections are all equal: formula_255.)
Quotient groups.
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 normal. Given any normal subgroup "N", the quotient group is defined by
formula_256
where the notation formula_257 is read as "formula_28 modulo formula_259".}} This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group formula_28: the product of two cosets formula_261 and formula_262 is formula_263 for all formula_186 and formula_189 in formula_28. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map formula_267 that associates to any element formula_186 its coset formula_261 should be a group homomorphism, or by general abstract considerations called universal properties. The coset formula_270 serves as the identity in this group, and the inverse of formula_261 in the quotient group is 
The elements of the quotient group formula_272 are formula_245 itself, which represents the identity, and formula_274. The group operation on the quotient is shown in the table. For example, formula_275. Both the subgroup formula_212, as well as the corresponding quotient are abelian, whereas formula_92 is not abelian. Building bigger groups by smaller ones, such as formula_92 from its subgroup formula_245 and the quotient formula_272 is abstracted by a notion called semidirect product.
Quotient groups and subgroups together form a way of describing every group by its "presentation": any group is the quotient of the free group over the "generators" of the group, quotiented by the subgroup of "relations". The dihedral group formula_92, for example, can be generated by two elements formula_282 and formula_137 (for example, formula_284, the right rotation and formula_285 the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations
formula_286
the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups.
Sub- and quotient groups are related in the following way: a subgroup formula_190 of formula_28 corresponds to an injective map formula_289, for which any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions. In general, homomorphisms are neither injective nor surjective. The kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.
Examples and applications.
Examples and applications of groups abound. A starting point is the group formula_6 of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk 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 winding once 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, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.
In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science 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 rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.
Integers.
The group of integers formula_6 under addition, denoted formula_292, has been described above. The integers, with the operation of multiplication instead of addition, formula_293 do "not" form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, formula_294 is an integer, but the only solution to the equation formula_295 in this case is formula_296, which is a rational number, but not an integer. Hence not every element of formula_6 has a (multiplicative) inverse.
Rationals.
The desire for the existence of multiplicative inverses suggests considering fractions
formula_298
Fractions of integers (with formula_3 nonzero) are known as rational numbers. The set of all such irreducible fractions is commonly denoted formula_300. There is still a minor obstacle for formula_301, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no formula_70 such that formula_303), formula_301 is still not a group.
However, the set of all "nonzero" rational numbers formula_305 does form an abelian group under multiplication, also denoted 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. Finally, the inverse of formula_306 is formula_307, therefore the axiom of the inverse element is satisfied.
The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in formula_300 – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
Modular arithmetic.
Modular arithmetic for a "modulus" formula_309 defines any two elements formula_2 and formula_3 that differ by a multiple of formula_309 to be equivalent, denoted by formula_313. Every integer is equivalent to one of the integers from formula_69 to formula_315, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from formula_69 to formula_315, forms a group, denoted as formula_318 or formula_319, with formula_69 as the identity element and formula_321 as the inverse element of formula_2.
A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on formula_323 and is advanced formula_324 hours, it ends up on formula_72, as shown in the illustration. This is expressed by saying that formula_326 is congruent to formula_72 "modulo formula_328" or, in symbols,
formula_329
For any prime number formula_330, there is also the multiplicative group of integers modulo formula_330. Its elements can be represented by formula_72 to formula_333. The group operation, multiplication modulo formula_330, replaces the usual product by its representative, the remainder of division by formula_330. For example, for formula_336, the four group elements can be represented by formula_337. In this group, formula_338, because the usual product formula_339 is equivalent to formula_72: when divided by formula_341 it yields a remainder of formula_72. The primality of formula_330 ensures that the usual product of two representatives is not divisible by formula_330, and therefore that the modular product is nonzero. The identity element is represented and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer formula_2 not divisible by formula_330, there exists an integer formula_3 such that 
formula_348
that is, such that formula_330 evenly divides formula_350. The inverse formula_3 can be found by using Bézout's identity and the fact that the greatest common divisor formula_352 In the case formula_336 above, the inverse of the element represented by formula_324 is that represented by formula_324, and the inverse of the element represented by formula_356 is represented , as formula_357. Hence all group axioms are fulfilled. This example is similar to formula_358 above: it consists of exactly those elements in the ring formula_359 that have a multiplicative inverse. These groups, denoted formula_360, are crucial to public-key cryptography.
Cyclic groups.
A "cyclic group" is a group all of whose elements are powers of a particular element formula_2. In multiplicative notation, the elements of the group are
formula_362
where formula_363 means formula_364, formula_365 stands for formula_366, etc. Such an element formula_2 is called a generator or a primitive element 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
formula_368
In the groups formula_319 introduced above, the element formula_72 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are formula_72. Any cyclic group with formula_309 elements is isomorphic to this group. A second example for cyclic groups is the group of formula_309th complex roots of unity, given by complex numbers formula_374 satisfying formula_375. These numbers can be visualized as the vertices on a regular formula_309-gon, as shown in blue in the image for formula_377. The group operation is multiplication of complex numbers. In the picture, multiplying with formula_374 corresponds to a counter-clockwise rotation by 60°. From field theory, the group formula_360 is cyclic for prime formula_330: for example, if formula_336, formula_356 is a generator since formula_383, formula_384, formula_385, and formula_386.
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element formula_2, all the powers of formula_2 are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to formula_389, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.
The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.
Symmetry groups.
"Symmetry groups" are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object "X" if every group element can be associated to some operation on "X" and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.
In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical 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 Jahn–Teller effect 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 phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
General linear group and representation theory.
Matrix groups consist of matrices together with matrix multiplication. The "general linear group" formula_390 consists of all invertible formula_309-by-formula_309 matrices with real entries. Its subgroups are referred to as "matrix groups" or "linear groups". The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group formula_393. It describes all possible rotations in formula_309 dimensions. Rotation matrices in this group are used in computer graphics.
"Representation theory" is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations in which the group acts on a vector space, such as the three-dimensional Euclidean space formula_395. A representation of a group formula_28 on an formula_309-dimensional real vector space is simply a group homomorphism
formula_398
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
A group action 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, algebraic groups and topological groups, especially (locally) compact groups.
Galois groups.
"Galois groups" were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation formula_399 are given by
formula_400
Each solution can be obtained by replacing the formula_401 sign by formula_5 or formula_403; analogous formula are known for cubic and quartic equations, but do "not" exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.
Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
Finite groups.
A group is called "finite" if it has a finite number of elements. The number of elements is called the order of the group. An important class is the "symmetric groups" formula_404, the groups of permutations of formula_259 objects. For example, the symmetric group on 3 letters formula_406 is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group formula_404 for a suitable integer formula_259, according to Cayley's theorem. Parallel to the group of symmetries of the square above, formula_406 can also be interpreted as the group of symmetries of an equilateral triangle.
The order of an element formula_2 in a group formula_28 is the least positive integer formula_309 such that formula_413, where formula_414 represents
formula_415
that is, application of the operation "formula_30" to formula_309 copies of formula_2. (If "formula_30" represents multiplication, then formula_414 corresponds to the formula_309th power of formula_2.) In infinite groups, such an formula_309 may not exist, in which case the order of formula_2 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: Lagrange's Theorem states that for a finite group formula_28 the order of any finite subgroup formula_190 divides the order of formula_28. The Sylow theorems give a partial converse.
The dihedral group formula_92 of symmetries of a square is a finite group of order 8. In this group, the order of formula_83 is 4, as is the order of the subgroup formula_245 that this element generates. The order of the reflection elements formula_86 etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups formula_360 of multiplication modulo a prime formula_330 have order formula_333.
Classification of finite simple groups.
Mathematicians often strive for a complete classification (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 formula_330, a prime number, are necessarily cyclic groups formula_436 and hence also abelian. Groups of order formula_437 can also be shown to be abelian, a statement which does not generalize to order formula_438, as the non-abelian group formula_92 of order formula_440 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called "simple" if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group (the "monster group") and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
Groups with additional structure.
An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set formula_28 equipped with a binary operation formula_442 (the group operation), a unary operation formula_443 (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs. 
This way of defining groups lends itself to generalizations such as the notion of a group objects in a category. Briefly this is an object (that is, examples of another mathematical structure) which comes with transformations (called morphisms) that mimic the group axioms. 
Topological groups.
Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, formula_444 and formula_445 must not vary wildly if formula_186 and formula_189 vary only a little. Such groups are called "topological groups," and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of -adic numbers. These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.
Lie groups.
A "Lie group" is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.
A standard example is the general linear group introduced above: it is an open subset of the space of all formula_309-by-formula_309 matrices, because it is given by the inequality
formula_450
where formula_451 denotes an formula_309-by-formula_309 matrix.
Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. 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 is the group of Lorentz transformations, 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 Minkowski space. The latter serves—in the absence of significant gravitation—as a model of spacetime in special relativity. The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.
Generalizations.
In abstract algebra, 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 monoid. The natural numbers formula_454 (including zero) under addition form a monoid, as do the nonzero integers under multiplication formula_455, see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as formula_456 is derived from formula_455, known as the Grothendieck group.
Groupoids are similar to groups except that the composition formula_34 need not be defined for all formula_2 and formula_3. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary -ary one (i.e., an operation taking arguments). With the proper generalization of the group axioms this gives rise to an -ary group The table gives a list of several structures generalizing groups.