In mathematics, a group is an algebraic structure consisting of a set
of elements equipped with an operation that combines any two elements
to form a third element and that satisfies four conditions called the
group axioms, namely closure, associativity, identity and
invertibility. One of the most familiar examples of a group is the set
of integers together with the addition operation, but the abstract
formalization of the group axioms, detached as it is from the concrete
nature of any particular group and its operation, applies much more
widely. It allows entities with highly diverse mathematical origins in
abstract algebra and beyond to be handled in a flexible way while
retaining their essential structural aspects. The ubiquity of groups
in numerous areas 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 are the
symmetry groups used in the
Algebraic structure → Group theory Group theory
Subgroup Normal subgroup
Quotient group (Semi-)direct product
kernel image direct sum
wreath product simple finite
infinite continuous multiplicative
additive cyclic abelian dihedral
List of group theory topics
Classification of finite simple groups
cyclic alternating Lie type sporadic
Sylow theorems Hall's theorem
p-group Elementary abelian group
Discrete groups Lattices
Arithmetic group Lattice Hyperbolic group
Topological / Lie groups
General linear GL(n)
G2 F4 E6 E7 E8
Lorentz Poincaré Conformal
Infinite dimensional Lie group
O(∞) SU(∞) Sp(∞)
Linear algebraic group
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Semigroup / Monoid
Racks and quandles
Abelian group Magma Lie group
Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra
Map of lattices Lattice theory
Module Group with operators Vector space
Associative Non-associative Composition algebra
Lie algebra Graded Bialgebra
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1 Definition and illustration
1.1 First example: the integers 1.2 Definition 1.3 Second example: a symmetry group
2 History 3 Elementary consequences of the group axioms
3.1 Uniqueness of identity element and inverses 3.2 Division
4 Basic concepts
4.1 Group homomorphisms 4.2 Subgroups 4.3 Cosets 4.4 Quotient groups
5 Examples and applications
5.1.1 Integers 5.1.2 Rationals
6 Finite groups
6.1 Classification of finite simple groups
7 Groups with additional structure
7.1 Topological groups 7.2 Lie groups
8 Generalizations 9 See also 10 Notes 11 Citations 12 References
12.1 General references
Definition and illustration First example: the integers One of the most familiar groups is the set of integers Z which consists of the numbers
..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ..., together with addition.
The following properties of integer addition serve as a model for the abstract group axioms given in the definition below.
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.
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
If a is any integer, then 0 + a = a + 0 = a.
The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following abstract definition is developed. Definition
The axioms for a group are short and natural... Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.
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. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:
Closure For all a, b in G, the result of the operation, a • b, is also in G.b[›] Associativity For all a, b and c in G, (a • b) • c = a • (b • c). Identity element There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element. Inverse element For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element.
The result of an operation may depend on the order of the operands. 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
a • b = b • a
may not always be true. This equation always holds in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). Groups for which the commutativity equation a • b = b • a always holds are called abelian groups (in honor of Niels Henrik Abel). The symmetry group described in the following section is an example of a group that is not abelian. The identity element of a group G is often written as 1 or 1G, a notation inherited from the multiplicative identity. 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. The identity element can also be written as id. The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). 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, •)". Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set. 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:
The elements of the symmetry group of the square (D4). Vertices are identified by color or number.
id (keeping it as it is)
r1 (rotation by 90° clockwise)
r2 (rotation by 180° clockwise)
r3 (rotation by 270° clockwise)
fv (vertical reflection)
fh (horizontal reflection)
fd (diagonal reflection)
fc (counter-diagonal reflection)
the identity operation leaving everything unchanged, denoted id; rotations of the square around its center by 90° clockwise, 180° clockwise, and 270° clockwise, denoted by r1, r2 and r3, respectively; reflections about the vertical and horizontal middle line (fh and fv), or through the two diagonals (fd and fc).
These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. 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. Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4 and denoted D4. The underlying set of the group is the above set of symmetry functions, 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 a and then b is written symbolically from right to left as
b • a ("apply the symmetry b after performing the symmetry a").
The right-to-left notation is the same notation that is used for composition of functions. The group table on the right lists the results of all such compositions possible. For example, rotating by 270° clockwise (r3) and then reflecting horizontally (fh) is the same as performing a reflection along the diagonal (fd). Using the above symbols, highlighted in blue in the group table:
fh • r3 = fd.
Group table of D4
• id r1 r2 r3 fv fh fd fc
id id r1 r2 r3 fv fh fd fc
r1 r1 r2 r3 id fc fd fv fh
r2 r2 r3 id r1 fh fv fc fd
r3 r3 id r1 r2 fd fc fh fv
fv fv fd fh fc id r2 r1 r3
fh fh fc fv fd r2 id r3 r1
fd fd fh fc fv r3 r1 id r2
fc fc fv fd fh r1 r3 r2 id
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.
Given this set of symmetries and the described operation, the group axioms can be understood as follows:
The closure axiom demands that the composition b • a of any two symmetries a and b is also a symmetry. Another example for the group operation is
r3 • fh = fc,
i.e., rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (fc). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table. The associativity constraint deals with composing more than two symmetries: Starting with three elements a, b and c of D4, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose a and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with a. The associativity condition
(a • b) • c = a • (b • c)
means that these two ways are the same, i.e., a product of many group elements can be simplified in any grouping. For example, (fd • fv) • r2 = fd • (fv • r2) can be checked using the group table at the right
(fd • fv) • r2 = r3 • r2 = r1, which equals
fd • (fv • r2) = fd • fh = r1.
While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. For instance, subtraction of numbers is not associative: (7 − 3) − 2 = 2 is not the same as 7 − (3 − 2) = 6. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form,
id • a = a, a • id = a.
An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of the following transformations—identity id, the reflections fh, fv, fd, fc and the 180° rotation r2—is its own inverse, because performing it twice brings the square back to its original orientation. The rotations r3 and r1 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. In symbols,
fh • fh = id, r3 • r1 = r1 • r3 = id.
In contrast to the group of integers above, where the order of the
operation is irrelevant, it does matter in D4: fh • r1 = fc but r1
• fh = fd. In other words, D4 is not abelian, which makes the group
structure more difficult than the integers introduced first.
Main article: History of group theory
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
a • b • c = (a • b) • c = a • (b • c)
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. The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above. Uniqueness of identity element and inverses Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element. To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted b and c, in a group (G, •). Then
b = b • e as 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 to rearrange the parentheses
= e • c since b is an inverse of a, i.e., b • a = e
= c for e is the identity element
The term b on the first line above and the c on the last are equal, since they are connected by a chain of equalities. In other words, there is only one inverse element of a. Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements e and f. Then e = e • f = f, hence e and f are equal. Division 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−1. In fact, we have
(b • a−1) • a = b • (a−1 • a) = b • e = b.
Uniqueness results by multiplying the two sides of the equation x • a = b by a−1. The element b • a−1, often denoted b / a, is called the right quotient of b by a, or the result of the right division of b by a. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a−1 • b. This solution is the left quotient of b by a, and is sometimes denoted a b. 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. In this case, the group operation is often denoted as an addition, and one talks of subtraction and difference instead of division and quotient. A consequence of this is that multiplication by a group element g is a bijection. Specifically, if g is an element of the group G, the function (mathematics) from G to itself that maps h ∈ G to g • h is a bijection. This function is called the left translation by g . 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. Basic concepts
The following sections use mathematical symbols such as X = x, y, z to denote a set X containing elements x, y, and z, or alternatively x ∈ X to restate that x is an element of X. The notation f : X → Y means f is a function assigning to every element of X an element of Y.
Further information: Glossary of group theory To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups. Group homomorphisms Main article: Group homomorphism Group homomorphismsg[›] are functions that preserve group structure. A function a: G → H between two groups (G, •) and (H, ∗) is called a homomorphism if the equation
a(g • k) = a(g) ∗ a(k)
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. This requirement ensures that a(1G) = 1H, and also a(g)−1 = a(g−1) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms. 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. 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. Subgroups Main article: Subgroup 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 h1−1, so the elements of H, equipped with the group operation on G restricted to H, indeed form a group. 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). 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 g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›] Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S. 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. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup. Cosets Main article: Coset 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 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. 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
gH = g • h : h ∈ H and Hg = h • g : h ∈ H , respectively.
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. The first case g1H = g2H happens precisely when g1−1 • 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. If they are, i.e., for all g in G, gH = Hg, then H is said to be a normal subgroup. 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). The subgroup R is also normal, because fcR = U = Rfc and similarly for any element other than fc. (In fact, in the case of D4, observe that all such cosets are equal, such that fhR = fvR = fdR = fcR.) Quotient groups Main article: Quotient group 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
G / N = gN, g ∈ G , "G modulo N".
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. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N.e[›]
Group table of the quotient group D4 / R
• R U
R R U
U U R
The elements of the quotient group D4 / R are R itself, which represents the identity, and U = fvR. The group operation on the quotient is shown at the right. 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. 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. 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 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. Together with the relations
r 4 = f 2 = (r • f)2 = 1,
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 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. The counterpart to
injective maps are surjective maps (every element of the target is
mapped onto), such as the canonical map G → G / N.y[›]
Interpreting subgroup and quotients in light of these homomorphisms
emphasizes the structural concept inherent to these definitions
alluded to in the introduction. In general, homomorphisms are neither
injective nor surjective. Kernel and image of group homomorphisms and
the first isomorphism theorem address this phenomenon.
Examples and applications
Examples of groups
A periodic wallpaper pattern gives rise to a wallpaper group.
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers.
Examples and applications of groups abound. A starting point is the
group Z of integers with addition as group operation, introduced
above. If instead of addition 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.i[›] For example, elements of the fundamental group are
represented by loops. The second image at the right shows some loops
in a plane minus a point. The blue loop is considered 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.j[›] 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.
displaystyle frac a b .
Fractions of integers (with b nonzero) are known as rational
numbers.l[›] The set of all such fractions is commonly denoted Q.
There is still a minor obstacle for (Q, ·), the rationals with
multiplication, being a group: because the rational number 0 does not
have a multiplicative inverse (i.e., there is no x such that x · 0 =
1), (Q, ·) is still not a group.
However, the set of all nonzero rational numbers Q ∖ 0 = q ∈ Q
q ≠ 0 does form an abelian group under multiplication, denoted (Q
∖ 0 , ·).m[›]
The hours on a clock form a group that uses addition modulo 12. Here 9 + 4 = 1.
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. 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. 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. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
9 + 4 ≡ 1 modulo 12.
The group of integers modulo n is written Zn or Z/nZ. For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. 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. For example, if p = 5, there are four group elements 1, 2, 3, 4. 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
16 ≡ 1 (mod 5).
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.o[›] The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that
a · b ≡ 1 (mod p), i.e., p divides the difference a · b − 1.
The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor gcd(a, p) equals 1. In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to (Q ∖ 0 , ·) above: it consists of exactly those elements in Z/pZ that have a multiplicative inverse. These groups are denoted Fp×. They are crucial to public-key cryptography.p[›] Cyclic groups Main article: Cyclic group
The 6th complex roots of unity form a cyclic group. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2.
A cyclic group is a group all of whose elements are powers of a particular element a. In multiplicative notation, the elements of the group are:
..., a−3, a−2, a−1, a0 = e, a, a2, a3, ...,
where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1 = (a • a • a)−1 etc.h[›] Such an element a 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
..., −a−a, −a, 0, a, a+a, ...
In the groups Z/nZ introduced above, the element 1 is primitive, so
these groups are cyclic. Indeed, each element is expressible as a sum
all of whose terms are 1. Any cyclic group with n elements is
isomorphic to this group. A second example for cyclic groups is the
group of n-th complex roots of unity, given by complex numbers z
satisfying zn = 1. These numbers can be visualized as the vertices on
a regular n-gon, as shown in blue at the right for n = 6. The group
operation is multiplication of complex numbers. In the picture,
multiplying with z corresponds to a counter-clockwise rotation by
60°. Using some field theory, the group Fp× can be shown to be
cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9
≡ 4, 33 ≡ 2, and 34 ≡ 1.
Some cyclic groups have an infinite number of elements. In these
groups, for every non-zero element a, all the powers of a are
distinct; despite the name "cyclic group", the powers of the elements
do not cycle. An infinite cyclic group is isomorphic to (Z, +), the
group of integers under addition introduced above. As these two
prototypes are both abelian, so is any cyclic group.
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.
Rotations and reflections form the symmetry group of a great icosahedron.
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
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.u[›] Geometric
properties that remain stable under group actions are investigated in
(geometric) invariant theory.
General linear group
Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2.
Matrix groups consist of matrices together with matrix multiplication.
The general linear group GL(n, R) consists of all invertible n-by-n
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 SO(n). It describes all
possible rotations in n dimensions. Via Euler angles, rotation
matrices are used in computer graphics.
ρ: G → GL(n, R)
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.w[›] Given a group action, this gives further means to study the object being acted on.x[›] 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 Main article: Galois group Galois groups were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation ax2 + bx + c = 0 are given by
− b ±
− 4 a c
displaystyle x= frac -bpm sqrt b^ 2 -4ac 2a .
Exchanging "+" and "−" in the expression, i.e., permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. 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. The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. 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. Finite groups Main article: Finite group 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 SN, the groups of permutations of N letters. 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. 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. Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle. The order of an element a in a group G is the least positive integer n such that an = e, where an represents
a ⋯ a
displaystyle underbrace acdots a _ n text factors ,
i.e., application of the operation • to n copies of a. (If •
represents multiplication, then an corresponds to the nth power of a.)
In infinite groups, such an n may not exist, in which case the order
of a is said to be infinity. The order of an element equals the order
of the cyclic subgroup generated by this element.
More sophisticated counting techniques, for example counting cosets,
yield more precise statements about finite groups: Lagrange's Theorem
states that for a finite group G the order of any finite subgroup H
divides the order of G. The
The unit circle in the complex plane under complex multiplication is a
Main article: Topological group 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, that is, g • h, and g−1 must not vary wildly if g and h vary only 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 reals R under addition, (R ∖ 0 , ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:
f ( x )
d x =
f ( x + c )
displaystyle int _ mathbb R f(x),dx=int _ mathbb R f(x+c),dx
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. 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. An advanced generalization of this idea, adapted to
the needs of algebraic geometry, is the étale fundamental group.
Main article: Lie group
Lie groups (in honor of Sophus Lie) are groups which also have a
manifold structure, i.e., they are spaces looking locally like some
det (A) ≠ 0,
where A denotes an n-by-n 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.v[›] Another example are the 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 space time 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. Generalizations
Totalityα Associativity Identity Invertibility Commutativity
Semigroupoid Unneeded Required Unneeded Unneeded Unneeded
Category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Magma Required Unneeded Unneeded Unneeded Unneeded
Quasigroup Required Unneeded Unneeded Required Unneeded
Loop Required Unneeded Required Required Unneeded
Semigroup Required Required Unneeded Unneeded Unneeded
Monoid Required Required Required Unneeded Unneeded
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.
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 N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ 0 , ·), see above. 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. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. 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 n-ary one (i.e., an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.
Finitely presented group
List of small groups
Mathematical Reviews lists 3,224 research papers on
group theory and its generalizations written in 2005.
^ aa: The classification was announced in 1983, but gaps
were found in the proof. See classification of finite simple groups
for further information.
^ b: The closure axiom is already implied by the condition
that • be a binary operation. Some authors therefore omit this
axiom. However, group constructions often start with an operation
defined on a superset, so a closure step is common in proofs that a
system is a group. Lang 2002
^ c: See, for example, the books of Lang (2002, 2005) and
Herstein (1996, 1975).
^ d: However, a group is not determined by its lattice of
subgroups. See Suzuki 1951.
^ e: The fact that the group operation extends this
canonically is an instance of a universal property.
^ f: For example, if G is finite, then the size of any
subgroup and any quotient group divides the size of G, according to
^ g: The word homomorphism derives from Greek
ὁμός—the same and μορφή—structure.
^ h: The additive notation for elements of a cyclic group
would be t • a, t in Z.
^ i: See the
Seifert–van Kampen theorem
^ Herstein 1975, §2, p. 26 ^ Hall 1967, §1.1, p. 1: "The idea of a group is one which pervades the whole of mathematics both pure and applied." ^ Lang 2005, App. 2, p. 360 ^ Cook, Mariana R. (2009), Mathematicians: An Outer View of the Inner World, Princeton, N.J.: Princeton University Press, p. 24, ISBN 9780691139517 ^ Herstein 1975, §2.1, p. 27 ^ Weisstein, Eric W. "Identity Element". MathWorld. ^ Herstein 1975, §2.6, p. 54 ^ Wussing 2007 ^ Kleiner 1986 ^ Smith 1906 ^ Galois 1908 ^ Kleiner 1986, p. 202 ^ Cayley 1889 ^ Wussing 2007, §III.2 ^ Lie 1973 ^ Kleiner 1986, p. 204 ^ Wussing 2007, §I.3.4 ^ Jordan 1870 ^ von Dyck 1882 ^ Curtis 2003 ^ Mackey 1976 ^ Borel 2001 ^ Aschbacher 2004 ^ Ledermann 1953, §1.2, pp. 4–5 ^ Ledermann 1973, §I.1, p. 3 ^ Lang 2002, §I.2, p. 7 ^ a b Lang 2005, §II.1, p. 17 ^ a b Mac Lane 1998 ^ Lang 2005, §II.3, p. 34 ^ Lang 2005, §II.1, p. 19 ^ Ledermann 1973, §II.12, p. 39 ^ Lang 2005, §II.4, p. 41 ^ Lang 2002, §I.2, p. 12 ^ Lang 2005, §II.4, p. 45 ^ Lang 2002, §I.2, p. 9 ^ Hatcher 2002, Chapter I, p. 30 ^ Coornaert, Delzant & Papadopoulos 1990 ^ for example, class groups and Picard groups; see Neukirch 1999, in particular §§I.12 and I.13 ^ Seress 1997 ^ Lang 2005, Chapter VII ^ Rosen 2000, p. 54 (Theorem 2.1) ^ Lang 2005, §VIII.1, p. 292 ^ Lang 2005, §II.1, p. 22 ^ Lang 2005, §II.2, p. 26 ^ Lang 2005, §II.1, p. 22 (example 11) ^ Lang 2002, §I.5, p. 26, 29 ^ Weyl 1952 ^ Conway, Delgado Friedrichs & Huson et al. 2001. See also Bishop 1993 ^ Bersuker, Isaac (2006), The Jahn-Teller Effect, Cambridge University Press, p. 2, ISBN 0-521-82212-2 ^ Jahn & Teller 1937 ^ Dove, Martin T (2003), Structure and Dynamics: an atomic view of materials, Oxford University Press, p. 265, ISBN 0-19-850678-3 ^ Welsh 1989 ^ Mumford, Fogarty & Kirwan 1994 ^ Lay 2003 ^ Kuipers 1999 ^ a b Fulton & Harris 1991 ^ Serre 1977 ^ Rudin 1990 ^ Robinson 1996, p. viii ^ Artin 1998 ^ Lang 2002, Chapter VI (see in particular p. 273 for concrete examples) ^ Lang 2002, p. 292 (Theorem VI.7.2) ^ Kurzweil & Stellmacher 2004 ^ Artin 1991, Theorem 6.1.14. See also Lang 2002, p. 77 for similar results. ^ Lang 2002, §I. 3, p. 22 ^ Ronan 2007 ^ Husain 1966 ^ Neukirch 1999 ^ Shatz 1972 ^ Milne 1980 ^ Warner 1983 ^ Borel 1991 ^ Goldstein 1980 ^ Weinberg 1972 ^ Naber 2003 ^ Becchi 1997 ^ Denecke & Wismath 2002 ^ Romanowska & Smith 2002 ^ Dudek 2001
References General references
Artin, Michael (1991), Algebra, Prentice Hall,
ISBN 978-0-89871-510-1 , Chapter 2 contains an
undergraduate-level exposition of the notions covered in this article.
Devlin, Keith (2000), The Language of Mathematics: Making the
Invisible Visible, Owl Books, ISBN 978-0-8050-7254-9 ,
Chapter 5 provides a layman-accessible explanation of groups.
Hall, G. G. (1967), Applied group theory, American Elsevier Publishing
Co., Inc., New York, MR 0219593 , an elementary
Herstein, Israel Nathan (1996),
Artin, Emil (1998), Galois Theory, New York: Dover Publications,
ISBN 978-0-486-62342-9 .
Aschbacher, Michael (2004), "The Status of the Classification of the
Finite Simple Groups" (PDF), Notices of the American Mathematical
Society, 51 (7): 736–740 .
Becchi, C. (1997), Introduction to Gauge Theories, p. 5211,
arXiv:hep-ph/9705211 , Bibcode:1997hep.ph....5211B .
Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), "The groups
of order at most 2000", Electronic Research Announcements of the
American Mathematical Society, 7: 1–4,
doi:10.1090/S1079-6762-01-00087-7, MR 1826989 .
Bishop, David H. L. (1993),
Historical references See also: Historically important publications in group theory
Borel, Armand (2001), Essays in the History of Lie Groups and
Algebraic Groups, Providence, R.I.: American Mathematical Society,
Cayley, Arthur (1889), The collected mathematical papers of Arthur
Cayley, II (1851–1860), Cambridge University Press .
O'Connor, John J.; Robertson, Edmund F., "The development of group
theory", MacTutor History of
v t e
Subgroup Normal subgroup Commutator subgroup Quotient group Group homomorphism (Semi-) direct product direct sum
Types of groups
General linear group
Exceptional Lie groups G2 F4 E6 E7 E8
Circle group Lorentz group Poincaré group Quaternion group
Infinite dimensional groups
History Applications Abst