Group (mathematics)

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This article is about basic notions of groups in mathematics. Group (mathematics)_sentence_0

For a more advanced treatment, see Group theory. Group (mathematics)_sentence_1

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. Group (mathematics)_sentence_2

One of the most familiar examples of a group is the set of integers together with the addition 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. Group (mathematics)_sentence_3

Groups share a fundamental kinship with the notion of symmetry. Group (mathematics)_sentence_4

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. Group (mathematics)_sentence_5

Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry. Group (mathematics)_sentence_6

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 (groupe, in French) for the symmetry group of the roots of an equation, now called a Galois group. Group (mathematics)_sentence_7

After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Group (mathematics)_sentence_8

Modern group theory—an active mathematical discipline—studies groups in their own right. Group (mathematics)_sentence_9

To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. Group (mathematics)_sentence_10

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. Group (mathematics)_sentence_11

A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Group (mathematics)_sentence_12

Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory. Group (mathematics)_sentence_13

Definition and illustration Group (mathematics)_section_0

First example: the integers Group (mathematics)_section_1

Group (mathematics)_description_list_0

  • ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ..., together with addition.Group (mathematics)_item_0_0

The following properties of integer addition serve as a model for the group axioms given in the definition below. Group (mathematics)_sentence_14

Group (mathematics)_unordered_list_1

  • For any two integers a and b, the sum a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition.Group (mathematics)_item_1_1
  • For all integers a, b and c, (a + b) + c = a + (b + c). 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 associativity.Group (mathematics)_item_1_2
  • If a is any integer, then 0 + a = a and a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.Group (mathematics)_item_1_3
  • For every integer a, there is an integer b such that a + b = 0 and b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a.Group (mathematics)_item_1_4

The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. Group (mathematics)_sentence_15

To appropriately understand these structures as a collective, the following definition is developed. Group (mathematics)_sentence_16

Definition Group (mathematics)_section_2

A group is a set, G, together with an operation ⋅ (called the group law of G) that combines any two elements a and b to form another element, denoted a ⋅ b or ab. Group (mathematics)_sentence_17

To qualify as a group, the set and operation, (G, ⋅), must satisfy four requirements known as the group axioms: Group (mathematics)_sentence_18

Group (mathematics)_description_list_2

  • Closure: For all a, b in G, the result of the operation, a ⋅ b, is also in G.Group (mathematics)_item_2_5
  • Associativity: For all a, b, c in G, one has (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).Group (mathematics)_item_2_6
  • Identity element: There exists an element e in G such that, for every a in G, the equations e ⋅ a = a and a ⋅ e = a hold. Such an element is unique (see below), and thus one speaks of the identity element.Group (mathematics)_item_2_7
  • Inverse element: For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. For each a, the b is unique and it is commonly denoted a (or −a, if the operation is denoted "+").Group (mathematics)_item_2_8

The result of the group operation may depend on the order of the operands. Group (mathematics)_sentence_19

In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation Group (mathematics)_sentence_20

Group (mathematics)_description_list_3

  • a ⋅ b = b ⋅ aGroup (mathematics)_item_3_9

may not be true for every two elements a and b. Group (mathematics)_sentence_21

This equation always holds in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). Group (mathematics)_sentence_22

Groups for which the commutativity equation a ⋅ b = b ⋅ a always holds are called abelian groups (in honor of Niels Henrik Abel). Group (mathematics)_sentence_23

The symmetry group described in the following section is an example of a group that is not abelian. Group (mathematics)_sentence_24

The identity element of a group G is often written as 1 or 1G, a notation inherited from the multiplicative identity. Group (mathematics)_sentence_25

If a group is abelian, then one may choose to denote the group operation by + and the identity element by 0; in that case, the group is called an additive group. Group (mathematics)_sentence_26

The identity element can also be written as id. Group (mathematics)_sentence_27

The set G is called the underlying set of the group (G, ⋅). Group (mathematics)_sentence_28

Often the group's underlying set G is used as a short name for the group (G, ⋅). Group (mathematics)_sentence_29

Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group (G, ⋅)" or "an element of the underlying set G of the group (G, ⋅)". Group (mathematics)_sentence_30

Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set. Group (mathematics)_sentence_31

As this formulation of the definition avoids existential quantifiers, it is generally preferred for computing with groups and for computer-aided proofs. Group (mathematics)_sentence_32

This formulation exhibits groups as a variety of universal algebra. Group (mathematics)_sentence_33

It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects. Group (mathematics)_sentence_34

Second example: a symmetry group Group (mathematics)_section_3

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Group (mathematics)_sentence_35

Any figure is congruent to itself. Group (mathematics)_sentence_36

However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. Group (mathematics)_sentence_37

A square has eight symmetries. Group (mathematics)_sentence_38

These are: Group (mathematics)_sentence_39

Group (mathematics)_table_general_0

The elements of the symmetry group of the square (D4). Vertices are identified by color or number.Group (mathematics)_table_caption_0
id (keeping it as it is)Group (mathematics)_cell_0_0_0 r1 (rotation by 90° clockwise)Group (mathematics)_cell_0_0_1 r2 (rotation by 180°)Group (mathematics)_cell_0_0_2 r3 (rotation by 270° clockwise)Group (mathematics)_cell_0_0_3
fv (vertical reflection)Group (mathematics)_cell_0_1_0 fh (horizontal reflection)Group (mathematics)_cell_0_1_1 fd (diagonal reflection)Group (mathematics)_cell_0_1_2 fc (counter-diagonal reflection)Group (mathematics)_cell_0_1_3

Group (mathematics)_unordered_list_4

  • the identity operation leaving everything unchanged, denoted id;Group (mathematics)_item_4_10
  • rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by r1, r2 and r3, respectively;Group (mathematics)_item_4_11
  • reflections about the horizontal and vertical middle line (fv and fh), or through the two diagonals (fd and fc).Group (mathematics)_item_4_12

These symmetries are functions. Group (mathematics)_sentence_40

Each sends a point in the square to the corresponding point under the symmetry. Group (mathematics)_sentence_41

For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Group (mathematics)_sentence_42

Composing two of these symmetries gives another symmetry. Group (mathematics)_sentence_43

These symmetries determine a group called the dihedral group of degree 4, denoted D4. Group (mathematics)_sentence_44

The underlying set of the group is the above set of symmetries, and the group operation is function composition. Group (mathematics)_sentence_45

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. Group (mathematics)_sentence_46

The result of performing first a and then b is written symbolically from right to left as b ° a ("apply the symmetry b after performing the symmetry a"). Group (mathematics)_sentence_47

(This is the usual notation for composition of functions.) Group (mathematics)_sentence_48

The group table on the right lists the results of all such compositions possible. Group (mathematics)_sentence_49

For example, rotating by 270° clockwise (r3) and then reflecting horizontally (fh) is the same as performing a reflection along the diagonal (fd). Group (mathematics)_sentence_50

Using the above symbols, highlighted in blue in the group table: Group (mathematics)_sentence_51

Group (mathematics)_description_list_5

  • fh ∘ r3 = fd.Group (mathematics)_item_5_13

Group (mathematics)_table_general_1

Group table of D4Group (mathematics)_table_caption_1
Group (mathematics)_header_cell_1_0_0 idGroup (mathematics)_header_cell_1_0_1 r1Group (mathematics)_header_cell_1_0_2 r2Group (mathematics)_header_cell_1_0_3 r3Group (mathematics)_header_cell_1_0_4 fvGroup (mathematics)_header_cell_1_0_5 fhGroup (mathematics)_header_cell_1_0_6 fdGroup (mathematics)_header_cell_1_0_7 fcGroup (mathematics)_header_cell_1_0_8
idGroup (mathematics)_header_cell_1_1_0 idGroup (mathematics)_cell_1_1_1 r1Group (mathematics)_cell_1_1_2 r2Group (mathematics)_cell_1_1_3 r3Group (mathematics)_cell_1_1_4 fvGroup (mathematics)_cell_1_1_5 fhGroup (mathematics)_cell_1_1_6 fdGroup (mathematics)_cell_1_1_7 fcGroup (mathematics)_cell_1_1_8
r1Group (mathematics)_header_cell_1_2_0 r1Group (mathematics)_cell_1_2_1 r2Group (mathematics)_cell_1_2_2 r3Group (mathematics)_cell_1_2_3 idGroup (mathematics)_cell_1_2_4 fcGroup (mathematics)_cell_1_2_5 fdGroup (mathematics)_cell_1_2_6 fvGroup (mathematics)_cell_1_2_7 fhGroup (mathematics)_cell_1_2_8
r2Group (mathematics)_header_cell_1_3_0 r2Group (mathematics)_cell_1_3_1 r3Group (mathematics)_cell_1_3_2 idGroup (mathematics)_cell_1_3_3 r1Group (mathematics)_cell_1_3_4 fhGroup (mathematics)_cell_1_3_5 fvGroup (mathematics)_cell_1_3_6 fcGroup (mathematics)_cell_1_3_7 fdGroup (mathematics)_cell_1_3_8
r3Group (mathematics)_header_cell_1_4_0 r3Group (mathematics)_cell_1_4_1 idGroup (mathematics)_cell_1_4_2 r1Group (mathematics)_cell_1_4_3 r2Group (mathematics)_cell_1_4_4 fdGroup (mathematics)_cell_1_4_5 fcGroup (mathematics)_cell_1_4_6 fhGroup (mathematics)_cell_1_4_7 fvGroup (mathematics)_cell_1_4_8
fvGroup (mathematics)_header_cell_1_5_0 fvGroup (mathematics)_cell_1_5_1 fdGroup (mathematics)_cell_1_5_2 fhGroup (mathematics)_cell_1_5_3 fcGroup (mathematics)_cell_1_5_4 idGroup (mathematics)_cell_1_5_5 r2Group (mathematics)_cell_1_5_6 r1Group (mathematics)_cell_1_5_7 r3Group (mathematics)_cell_1_5_8
fhGroup (mathematics)_header_cell_1_6_0 fhGroup (mathematics)_cell_1_6_1 fcGroup (mathematics)_cell_1_6_2 fvGroup (mathematics)_cell_1_6_3 fdGroup (mathematics)_cell_1_6_4 r2Group (mathematics)_cell_1_6_5 idGroup (mathematics)_cell_1_6_6 r3Group (mathematics)_cell_1_6_7 r1Group (mathematics)_cell_1_6_8
fdGroup (mathematics)_header_cell_1_7_0 fdGroup (mathematics)_cell_1_7_1 fhGroup (mathematics)_cell_1_7_2 fcGroup (mathematics)_cell_1_7_3 fvGroup (mathematics)_cell_1_7_4 r3Group (mathematics)_cell_1_7_5 r1Group (mathematics)_cell_1_7_6 idGroup (mathematics)_cell_1_7_7 r2Group (mathematics)_cell_1_7_8
fcGroup (mathematics)_header_cell_1_8_0 fcGroup (mathematics)_cell_1_8_1 fvGroup (mathematics)_cell_1_8_2 fdGroup (mathematics)_cell_1_8_3 fhGroup (mathematics)_cell_1_8_4 r1Group (mathematics)_cell_1_8_5 r3Group (mathematics)_cell_1_8_6 r2Group (mathematics)_cell_1_8_7 idGroup (mathematics)_cell_1_8_8
The elements id, r1, r2, and r3 form a subgroup, highlighted in   red (upper left region). A left and right coset of this subgroup is highlighted in   green (in the last row) and   yellow (last column), respectively.Group (mathematics)_cell_1_9_0

Given this set of symmetries and the described operation, the group axioms can be understood as follows: Group (mathematics)_sentence_52

In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in D4, as, for example, fh ∘ r1 = fc but r1 ∘ fh = fd. Group (mathematics)_sentence_53

In other words, D4 is not abelian, which makes the group structure more difficult than the integers introduced first. Group (mathematics)_sentence_54

History Group (mathematics)_section_4

Main article: History of group theory Group (mathematics)_sentence_55

The modern concept of an abstract group developed out of several fields of mathematics. Group (mathematics)_sentence_56

The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. Group (mathematics)_sentence_57

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). Group (mathematics)_sentence_58

The elements of such a Galois group correspond to certain permutations of the roots. Group (mathematics)_sentence_59

At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. Group (mathematics)_sentence_60

More general permutation groups were investigated in particular by Augustin Louis Cauchy. Group (mathematics)_sentence_61

Arthur Cayley's On the theory of groups, as depending on the symbolic equation θ = 1 (1854) gives the first abstract definition of a finite group. Group (mathematics)_sentence_62

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. Group (mathematics)_sentence_63

After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Group (mathematics)_sentence_64

Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884. Group (mathematics)_sentence_65

The third field contributing to group theory was number theory. Group (mathematics)_sentence_66

Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. Group (mathematics)_sentence_67

In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers. Group (mathematics)_sentence_68

The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Group (mathematics)_sentence_69

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. Group (mathematics)_sentence_70

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. Group (mathematics)_sentence_71

The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Group (mathematics)_sentence_72

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. Group (mathematics)_sentence_73

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. Group (mathematics)_sentence_74

This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Group (mathematics)_sentence_75

Research is ongoing to simplify the proof of this classification. Group (mathematics)_sentence_76

These days, group theory is still a highly active mathematical branch, impacting many other fields. Group (mathematics)_sentence_77

Elementary consequences of the group axioms Group (mathematics)_section_5

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. Group (mathematics)_sentence_78

For example, repeated applications of the associativity axiom show that the unambiguity of Group (mathematics)_sentence_79

Group (mathematics)_description_list_6

  • a ⋅ b ⋅ c = (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)Group (mathematics)_item_6_14

generalizes to more than three factors. Group (mathematics)_sentence_80

Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. Group (mathematics)_sentence_81

The axioms may be weakened to assert only the existence of a left identity and left inverses. Group (mathematics)_sentence_82

Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above. Group (mathematics)_sentence_83

Uniqueness of identity element Group (mathematics)_section_6

The group axioms imply that the identity element is unique: If e and f are identity elements of a group, then e = e ⋅ f = f. Group (mathematics)_sentence_84

Uniqueness of inverses Group (mathematics)_section_7

The group axioms also imply that the inverse of each element is unique: If a group element a has both b and c as inverses, then Group (mathematics)_sentence_85

Group (mathematics)_description_list_7

  • b = b ⋅ e      since e is the identity element = b ⋅ (a ⋅ c)      because c is an inverse of a, so e = a ⋅ c = (b ⋅ a) ⋅ c      by associativity, which allows rearranging the parentheses = e ⋅ c      since b is an inverse of a, i.e., b ⋅ a = e = c      since e is the identity element.Group (mathematics)_item_7_15

Thus, it is customary to speak of the identity, and the inverse of an element. Group (mathematics)_sentence_86

Division Group (mathematics)_section_8

In groups, the existence of inverse elements implies that division is possible: given elements a and b of the group G, there is exactly one solution x in G to the equation x ⋅ a = b, namely b ⋅ a. Group (mathematics)_sentence_87

In fact, we have Group (mathematics)_sentence_88

Group (mathematics)_description_list_8

  • (b ⋅ a) ⋅ a = b ⋅ (a ⋅ a) = b ⋅ e = b.Group (mathematics)_item_8_16

Uniqueness results by multiplying the two sides of the equation x ⋅ a = b by a. Group (mathematics)_sentence_89

The element b ⋅ a, often denoted b / a, is called the right quotient of b by a, or the result of the right division of b by a. Group (mathematics)_sentence_90

Similarly there is exactly one solution y in G to the equation a ⋅ y = b, namely y = a ⋅ b. Group (mathematics)_sentence_91

This solution is the left quotient of b by a, and is sometimes denoted a \ b. Group (mathematics)_sentence_92

In general b / a and a \ b may be different, but, if the group operation is commutative (that is, if the group is abelian), they are equal. Group (mathematics)_sentence_93

In this case, the group operation is often denoted as an addition, and one talks of subtraction and difference instead of division and quotient. Group (mathematics)_sentence_94

A consequence of this is that multiplication by a group element g is a bijection. Group (mathematics)_sentence_95

Specifically, if g is an element of the group G, the function from G to itself that maps h ∈ G to g ⋅ h is a bijection. Group (mathematics)_sentence_96

This function is called the left translation by g . Group (mathematics)_sentence_97

Similarly, the right translation by g is the bijection from G to itself, that maps h to h ⋅ g. If G is abelian, the left and the right translation by a group element are the same. Group (mathematics)_sentence_98

Basic concepts Group (mathematics)_section_9

Group (mathematics)_description_list_9

  • Group (mathematics)_item_9_17

To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. Group (mathematics)_sentence_99

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. Group (mathematics)_sentence_100

This compatibility manifests itself in the following notions in various ways. Group (mathematics)_sentence_101

For example, groups can be related to each other via functions called group homomorphisms. Group (mathematics)_sentence_102

By the mentioned principle, they are required to respect the group structures in a precise sense. Group (mathematics)_sentence_103

The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. Group (mathematics)_sentence_104

The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups. Group (mathematics)_sentence_105

Group homomorphisms Group (mathematics)_section_10

Main article: Group homomorphism Group (mathematics)_sentence_106

Group homomorphisms are functions that preserve group structure. Group (mathematics)_sentence_107

A function a: G → H between two groups (G, ⋅) and (H, ∗) is called a homomorphism if the equation Group (mathematics)_sentence_108

Group (mathematics)_description_list_10

  • a(g ⋅ k) = a(g) ∗ a(k)Group (mathematics)_item_10_18

holds for all elements g, k in G. In other words, the result is the same when performing the group operation after or before applying the map a. Group (mathematics)_sentence_109

This requirement ensures that a(1G) = 1H, and also a(g) = a(g) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms. Group (mathematics)_sentence_110

Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. Group (mathematics)_sentence_111

For example, proving that g ⋅ g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first. Group (mathematics)_sentence_112

Subgroups Group (mathematics)_section_11

Main article: Subgroup Group (mathematics)_sentence_113

Informally, a subgroup is a group H contained within a bigger one, G. Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are h1 ⋅ h2 and h1, so the elements of H, equipped with the group operation on G restricted to H, indeed form a group. Group (mathematics)_sentence_114

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: 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° (note that rotation in the opposite direction is not defined). Group (mathematics)_sentence_115

The subgroup test is a necessary and sufficient condition for a nonempty subset H of a group G to be a subgroup: it is sufficient to check that gh ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole. Group (mathematics)_sentence_116

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. Group (mathematics)_sentence_117

It is the smallest subgroup of G containing S. In the introductory example above, the subgroup generated by r2 and fv consists of these two elements, the identity element id and fh = fv ⋅ r2. Group (mathematics)_sentence_118

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. Group (mathematics)_sentence_119

Cosets Group (mathematics)_section_12

Main article: Coset Group (mathematics)_sentence_120

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. Group (mathematics)_sentence_121

For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e., the rotation operations are irrelevant to the question whether a reflection has been performed. Group (mathematics)_sentence_122

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. In symbolic terms, the left and right cosets of H containing g are Group (mathematics)_sentence_123

Group (mathematics)_description_list_11

  • gH = {g ⋅ h : h ∈ H} and Hg = {h ⋅ g : h ∈ H}, respectively.Group (mathematics)_item_11_19

The left cosets of any subgroup H form a partition of G; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection. Group (mathematics)_sentence_124

The first case g1H = g2H happens precisely when g1 ⋅ g2 ∈ H, i.e., 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. Group (mathematics)_sentence_125

If they are, i.e., for all g in G, gH = Hg, then H is said to be a normal subgroup. Group (mathematics)_sentence_126

In D4, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U = fcR = {fc, fv, fd, fh} (highlighted in green). Group (mathematics)_sentence_127

The subgroup R is also normal, because fcR = U = Rfc and similarly for any element other than fc. Group (mathematics)_sentence_128

(In fact, in the case of D4, observe that all such cosets are equal, such that fhR = fvR = fdR = fcR.) Group (mathematics)_sentence_129

Quotient groups Group (mathematics)_section_13

Main article: Quotient group Group (mathematics)_sentence_130

In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. Group (mathematics)_sentence_131

For this to be possible, the subgroup has to be normal. Group (mathematics)_sentence_132

Given any normal subgroup N, the quotient group is defined by Group (mathematics)_sentence_133

Group (mathematics)_description_list_12

  • G / N = {gN, g ∈ G}, "G modulo N".Group (mathematics)_item_12_20

This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) ⋅ (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. Group (mathematics)_sentence_134

The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN) = (g)N. Group (mathematics)_sentence_135

Group (mathematics)_table_general_2

Group table of the quotient group D4 / RGroup (mathematics)_table_caption_2
Group (mathematics)_header_cell_2_0_0 RGroup (mathematics)_header_cell_2_0_1 UGroup (mathematics)_header_cell_2_0_2
RGroup (mathematics)_header_cell_2_1_0 RGroup (mathematics)_cell_2_1_1 UGroup (mathematics)_cell_2_1_2
UGroup (mathematics)_header_cell_2_2_0 UGroup (mathematics)_cell_2_2_1 RGroup (mathematics)_cell_2_2_2

The elements of the quotient group D4 / R are R itself, which represents the identity, and U = fvR. Group (mathematics)_sentence_136

The group operation on the quotient is shown at the right. Group (mathematics)_sentence_137

For example, U ⋅ U = fvR ⋅ fvR = (fv ⋅ fv)R = R. Both the subgroup R = {id, r1, r2, r3}, as well as the corresponding quotient are abelian, whereas D4 is not abelian. Group (mathematics)_sentence_138

Building bigger groups by smaller ones, such as D4 from its subgroup R and the quotient D4 / R is abstracted by a notion called semidirect product. Group (mathematics)_sentence_139

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. Group (mathematics)_sentence_140

The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv 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. Group (mathematics)_sentence_141

Together with the relations Group (mathematics)_sentence_142

Group (mathematics)_description_list_13

  • r  = f  = (r ⋅ f) = 1,Group (mathematics)_item_13_21

the group is completely described. Group (mathematics)_sentence_143

A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups. Group (mathematics)_sentence_144

Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e., any element of the target has at most one element that maps to it. Group (mathematics)_sentence_145

The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N. Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. Group (mathematics)_sentence_146

In general, homomorphisms are neither injective nor surjective. Group (mathematics)_sentence_147

Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon. Group (mathematics)_sentence_148

Examples and applications Group (mathematics)_section_14

Main articles: Examples of groups and Applications of group theory Group (mathematics)_sentence_149

Examples and applications of groups abound. Group (mathematics)_sentence_150

A starting point is the group Z of integers with addition as group operation, introduced above. Group (mathematics)_sentence_151

If instead of addition multiplication is considered, one obtains multiplicative groups. Group (mathematics)_sentence_152

These groups are predecessors of important constructions in abstract algebra. Group (mathematics)_sentence_153

Groups are also applied in many other mathematical areas. Group (mathematics)_sentence_154

Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. Group (mathematics)_sentence_155

For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. Group (mathematics)_sentence_156

By means of this connection, topological properties such as proximity and continuity translate into properties of groups. Group (mathematics)_sentence_157

For example, elements of the fundamental group are represented by loops. Group (mathematics)_sentence_158

The second image at the right shows some loops in a plane minus a point. Group (mathematics)_sentence_159

The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. Group (mathematics)_sentence_160

The presence of the hole prevents the orange loop from being shrunk to a point. Group (mathematics)_sentence_161

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). Group (mathematics)_sentence_162

This way, the fundamental group detects the hole. Group (mathematics)_sentence_163

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. Group (mathematics)_sentence_164

In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Group (mathematics)_sentence_165

Further branches crucially applying groups include algebraic geometry and number theory. Group (mathematics)_sentence_166

In addition to the above theoretical applications, many practical applications of groups exist. Group (mathematics)_sentence_167

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. Group (mathematics)_sentence_168

Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept. Group (mathematics)_sentence_169

Numbers Group (mathematics)_section_15

Many number systems, such as the integers and the rationals enjoy a naturally given group structure. Group (mathematics)_sentence_170

In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Group (mathematics)_sentence_171

Such number systems are predecessors to more general algebraic structures known as rings and fields. Group (mathematics)_sentence_172

Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups. Group (mathematics)_sentence_173

Integers Group (mathematics)_section_16

Rationals Group (mathematics)_section_17

The desire for the existence of multiplicative inverses suggests considering fractions Group (mathematics)_sentence_174

Modular arithmetic Group (mathematics)_section_18

In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. Group (mathematics)_sentence_175

The result of modular addition is the remainder of that division. Group (mathematics)_sentence_176

For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. Group (mathematics)_sentence_177

This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. Group (mathematics)_sentence_178

This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols, Group (mathematics)_sentence_179

Group (mathematics)_description_list_14

  • 9 + 4 ≡ 1 modulo 12.Group (mathematics)_item_14_22

For any prime number p, there is also the multiplicative group of integers modulo p. Group (mathematics)_sentence_180

Its elements are the integers 1 to p − 1. Group (mathematics)_sentence_181

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. Group (mathematics)_sentence_182

For example, if p = 5, there are four group elements 1, 2, 3, 4. Group (mathematics)_sentence_183

In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted Group (mathematics)_sentence_184

Group (mathematics)_description_list_15

  • 16 ≡ 1 (mod 5).Group (mathematics)_item_15_23

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. Group (mathematics)_sentence_185

The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Group (mathematics)_sentence_186

Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that Group (mathematics)_sentence_187

Group (mathematics)_description_list_16

  • a · b ≡ 1 (mod p), i.e., p divides the difference a · b − 1.Group (mathematics)_item_16_24

Cyclic groups Group (mathematics)_section_19

Main article: Cyclic group Group (mathematics)_sentence_188

A cyclic group is a group all of whose elements are powers of a particular element a. Group (mathematics)_sentence_189

In multiplicative notation, the elements of the group are: Group (mathematics)_sentence_190

Group (mathematics)_description_list_17

  • ..., a, a, a, a = e, a, a, a, ...,Group (mathematics)_item_17_25

where a means a ⋅ a, and a stands for a ⋅ a ⋅ a = (a ⋅ a ⋅ a) etc. Group (mathematics)_sentence_191

Such an element a is called a generator or a primitive element of the group. Group (mathematics)_sentence_192

In additive notation, the requirement for an element to be primitive is that each element of the group can be written as Group (mathematics)_sentence_193

Group (mathematics)_description_list_18

  • ..., −a−a, −a, 0, a, a+a, ...Group (mathematics)_item_18_26

In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Group (mathematics)_sentence_194

Indeed, each element is expressible as a sum all of whose terms are 1. Group (mathematics)_sentence_195

Any cyclic group with n elements is isomorphic to this group. Group (mathematics)_sentence_196

A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying z = 1. Group (mathematics)_sentence_197

These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. Group (mathematics)_sentence_198

The group operation is multiplication of complex numbers. Group (mathematics)_sentence_199

In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Group (mathematics)_sentence_200

Using some field theory, the group Fp can be shown to be cyclic: for example, if p = 5, 3 is a generator since 3 = 3, 3 = 9 ≡ 4, 3 ≡ 2, and 3 ≡ 1. Group (mathematics)_sentence_201

Some cyclic groups have an infinite number of elements. Group (mathematics)_sentence_202

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. Group (mathematics)_sentence_203

An infinite cyclic group is isomorphic to (Z, +), the group of integers under addition introduced above. Group (mathematics)_sentence_204

As these two prototypes are both abelian, so is any cyclic group. Group (mathematics)_sentence_205

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. Group (mathematics)_sentence_206

Symmetry groups Group (mathematics)_section_20

Main article: Symmetry group Group (mathematics)_sentence_207

See also: Molecular symmetry, Space group, and Symmetry in physics Group (mathematics)_sentence_208

Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Group (mathematics)_sentence_209

Conceptually, group theory can be thought of as the study of symmetry. Group (mathematics)_sentence_210

Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. Group (mathematics)_sentence_211

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. Group (mathematics)_sentence_212

In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). Group (mathematics)_sentence_213

By a group action, the group pattern is connected to the structure of the object being acted on. Group (mathematics)_sentence_214

In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. Group (mathematics)_sentence_215

These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. Group (mathematics)_sentence_216

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. Group (mathematics)_sentence_217

Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. Group (mathematics)_sentence_218

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. Group (mathematics)_sentence_219

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. Group (mathematics)_sentence_220

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. Group (mathematics)_sentence_221

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons. Group (mathematics)_sentence_222

Group (mathematics)_table_general_3

Group (mathematics)_cell_3_0_0 Group (mathematics)_cell_3_0_1 Group (mathematics)_cell_3_0_2 Group (mathematics)_cell_3_0_3 Group (mathematics)_cell_3_0_4
Buckminsterfullerene displays

icosahedral symmetry, though the double bonds reduce this to pyritohedral symmetry.Group (mathematics)_cell_3_1_0

Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.Group (mathematics)_cell_3_1_1 Cubane C8H8 features
octahedral symmetry.Group (mathematics)_cell_3_1_2
Hexaaquacopper(II) complex ion, [Cu(OH2)6]. Compared to a perfectly symmetrical shape, the molecule is vertically dilated by about 22% (Jahn-Teller effect).Group (mathematics)_cell_3_1_3 The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.Group (mathematics)_cell_3_1_4

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. Group (mathematics)_sentence_223

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. Group (mathematics)_sentence_224

Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory. Group (mathematics)_sentence_225

General linear group and representation theory Group (mathematics)_section_21

Main articles: General linear group and Representation theory Group (mathematics)_sentence_226

Matrix groups consist of matrices together with matrix multiplication. Group (mathematics)_sentence_227

The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Group (mathematics)_sentence_228

Its subgroups are referred to as matrix groups or linear groups. Group (mathematics)_sentence_229

The dihedral group example mentioned above can be viewed as a (very small) matrix group. Group (mathematics)_sentence_230

Another important matrix group is the special orthogonal group SO(n). Group (mathematics)_sentence_231

It describes all possible rotations in n dimensions. Group (mathematics)_sentence_232

Via Euler angles, rotation matrices are used in computer graphics. Group (mathematics)_sentence_233

Representation theory is both an application of the group concept and important for a deeper understanding of groups. Group (mathematics)_sentence_234

It studies the group by its group actions on other spaces. Group (mathematics)_sentence_235

A broad class of group representations are linear representations, i.e., the group is acting on a vector space, such as the three-dimensional Euclidean space R. A representation of G on an n-dimensional real vector space is simply a group homomorphism Group (mathematics)_sentence_236

Group (mathematics)_description_list_19

  • ρ: G → GL(n, R)Group (mathematics)_item_19_27

from the group to the general linear group. Group (mathematics)_sentence_237

This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. Group (mathematics)_sentence_238

Given a group action, this gives further means to study the object being acted on. Group (mathematics)_sentence_239

On the other hand, it also yields information about the group. Group (mathematics)_sentence_240

Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups. Group (mathematics)_sentence_241

Galois groups Group (mathematics)_section_22

Main article: Galois group Group (mathematics)_sentence_242

Galois groups were developed to help solve polynomial equations by capturing their symmetry features. Group (mathematics)_sentence_243

For example, the solutions of the quadratic equation ax + bx + c = 0 are given by Group (mathematics)_sentence_244

Exchanging "+" and "−" in the expression, i.e., permuting the two solutions of the equation can be viewed as a (very simple) group operation. Group (mathematics)_sentence_245

Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Group (mathematics)_sentence_246

Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e., solutions expressible using solely addition, multiplication, and roots similar to the formula above. Group (mathematics)_sentence_247

The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Group (mathematics)_sentence_248

Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics. Group (mathematics)_sentence_249

Finite groups Group (mathematics)_section_23

Main article: Finite group Group (mathematics)_sentence_250

A group is called finite if it has a finite number of elements. Group (mathematics)_sentence_251

The number of elements is called the order of the group. Group (mathematics)_sentence_252

An important class is the symmetric groups SN, the groups of permutations of N letters. Group (mathematics)_sentence_253

For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e., contains the elements ABC, ACB, BAC, BCA, CAB, CBA, in total 6 (factorial of 3) elements. Group (mathematics)_sentence_254

This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N, according to Cayley's theorem. Group (mathematics)_sentence_255

Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle. Group (mathematics)_sentence_256

The order of an element a in a group G is the least positive integer n such that a = e, where a represents Group (mathematics)_sentence_257

i.e., application of the operation ⋅ to n copies of a. Group (mathematics)_sentence_258

(If ⋅ represents multiplication, then a corresponds to the nth power of a.) Group (mathematics)_sentence_259

In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. Group (mathematics)_sentence_260

The order of an element equals the order of the cyclic subgroup generated by this element. Group (mathematics)_sentence_261

More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G. The Sylow theorems give a partial converse. Group (mathematics)_sentence_262

The dihedral group (discussed above) is a finite group of order 8. Group (mathematics)_sentence_263

The order of r1 is 4, as is the order of the subgroup R it generates (see above). Group (mathematics)_sentence_264

The order of the reflection elements fv etc. is 2. Group (mathematics)_sentence_265

Both orders divide 8, as predicted by Lagrange's theorem. Group (mathematics)_sentence_266

The groups Fp above have order p − 1. Group (mathematics)_sentence_267

Classification of finite simple groups Group (mathematics)_section_24

Main article: Classification of finite simple groups Group (mathematics)_sentence_268

Mathematicians often strive for a complete classification (or list) of a mathematical notion. Group (mathematics)_sentence_269

In the context of finite groups, this aim leads to difficult mathematics. Group (mathematics)_sentence_270

According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Group (mathematics)_sentence_271

Groups of order p can also be shown to be abelian, a statement which does not generalize to order p, as the non-abelian group D4 of order 8 = 2 above shows. Group (mathematics)_sentence_272

Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. Group (mathematics)_sentence_273

An intermediate step is the classification of finite simple groups. Group (mathematics)_sentence_274

A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. Group (mathematics)_sentence_275

The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Group (mathematics)_sentence_276

Listing all finite simple groups was a major achievement in contemporary group theory. Group (mathematics)_sentence_277

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. Group (mathematics)_sentence_278

Groups with additional structure Group (mathematics)_section_25

Many groups are simultaneously groups and examples of other mathematical structures. Group (mathematics)_sentence_279

In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. Group (mathematics)_sentence_280

For example, every group (as defined above) is also a set, so a group is a group object in the category of sets. Group (mathematics)_sentence_281

Topological groups Group (mathematics)_section_26

Main article: Topological group Group (mathematics)_sentence_282

Some topological spaces may be endowed with a group law. Group (mathematics)_sentence_283

In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g ⋅ h, and g must not vary wildly if g and h vary only little. Group (mathematics)_sentence_284

Such groups are called topological groups, and they are the group objects in the category of topological spaces. Group (mathematics)_sentence_285

The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. Group (mathematics)_sentence_286

All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. Group (mathematics)_sentence_287

The former offer an abstract formalism of invariant integrals. Group (mathematics)_sentence_288

Invariance means, in the case of real numbers for example: Group (mathematics)_sentence_289

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Group (mathematics)_sentence_290

Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. Group (mathematics)_sentence_291

An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group. Group (mathematics)_sentence_292

Lie groups Group (mathematics)_section_27

Main article: Lie group Group (mathematics)_sentence_293

Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e., they are spaces looking locally like some Euclidean space of the appropriate dimension. Group (mathematics)_sentence_294

Again, the additional structure, here the manifold structure, has to be compatible, i.e., the maps corresponding to multiplication and the inverse have to be smooth. Group (mathematics)_sentence_295

A standard example is the general linear group introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality Group (mathematics)_sentence_296

Group (mathematics)_description_list_20

  • det (A) ≠ 0,Group (mathematics)_item_20_28

where A denotes an n-by-n matrix. Group (mathematics)_sentence_297

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Group (mathematics)_sentence_298

Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. Group (mathematics)_sentence_299

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. Group (mathematics)_sentence_300

Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. Group (mathematics)_sentence_301

They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. Group (mathematics)_sentence_302

The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. Group (mathematics)_sentence_303

The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. Group (mathematics)_sentence_304

By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Group (mathematics)_sentence_305

Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. Group (mathematics)_sentence_306

Generalizations Group (mathematics)_section_28

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. Group (mathematics)_sentence_307

For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. Group (mathematics)_sentence_308

The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. Group (mathematics)_sentence_309

There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Group (mathematics)_sentence_310

Groupoids are similar to groups except that the composition a ⋅ b need not be defined for all a and b. Group (mathematics)_sentence_311

They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Group (mathematics)_sentence_312

Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e., an operation taking n arguments). Group (mathematics)_sentence_313

With the proper generalization of the group axioms this gives rise to an n-ary group. Group (mathematics)_sentence_314

The table gives a list of several structures generalizing groups. Group (mathematics)_sentence_315

See also Group (mathematics)_section_29

Group (mathematics)_unordered_list_21

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: (mathematics).