Group (mathematics)

This article covers only some of the basic notions related to groups. Further ways of studying groups are treated in group theory.

The possible rearrangements of Rubik's Cube form a group, called the Rubik's Cube group.

In mathematics, a group is a set of elements with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must also satisfy a few technical conditions, such as associativity, and having both an identity element and inverse elements. A familiar example of a group is the set of integers and the operation addition "+"; the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, and every integer n has an additive inverse, −n. Because the definition of a group is so general, groups occur in many forms and have applications in numerous areas, both within and outside mathematics.

Groups share a fundamental kinship with the notion of symmetry. Any geometrical object possesses a group encoding its symmetry features, called its symmetry group. It consists of the set of transformations that leave the object unchanged; two such transformations are combined by performing one after the other, which always produces a third transformation that leaves the object unchanged. Such symmetry groups, particularly the continuous Lie groups, play an important role in in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity. They also find applications in chemistry.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830's. Getting input from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active part of the mathematical discipline of abstract algebra—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better understandable, pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Groups underlie many other algebraic objects such as rings and modules; they are a central organizing principle of algebra and contemporary mathematics in general.

Definition and illustration

A group is a set with an operation "•" that combines any two elements a and b to form another element denoted a • b. The abstract symbol "•" is to be understood as a general placeholder for a concretely given operation; for example, it might correspond to normal addition or multiplication, to a permutation of objects, or to a rotation of an object. To qualify as a group, the set and operation must satisfy four requirements known as the group axioms. Before giving these axioms, the concept of a group is illustrated with two examples: the integers, where the operation "•" corresponds to addition, and the symmetry group of the square, whose group operation "•" corresponds to rotations and flips that leave a square invariant.

First example: the integers

One of the most familiar groups is the set of integers Z containing the numbers

..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...

which are combined with the usual addition operation "+". The following properties of integer addition serve as a model for the more abstract group axioms given later.

1. For any two integers a and b, the sum a + b is also an integer. In other words, adding two integers does not leave the domain of integers Z, a property known as closure.
2. For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, if we first add a and b, and then add c to their sum, this gives the same final result as if we had added the sum of b and c to a. The two ways of calculating a + b + c give the same result, a property known as the associativity of the addition.
3. If a is any integer, then 0 + a = a + 0 = a. Adding 0 to any integer returns the same integer, which is why zero is called the identity element of addition.
4. For every integer a, there is another integer b = −a such that a + b = b + a = 0. −a is called the inverse element of the integer a.

Worked example: a symmetry group

id (keeping it as is) r1 (rotation by 90°) r2 (rotation by 180°) r3 (left rotation by 90°) fv (vertical flip) fh (horizontal flip) fd (diagonal flip) fc (counter-diagonal flip) Elements of the symmetry group. The vertices are colored only to visualize the operations.

group table • 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. A left and right coset of this subgroup is highlighted in green and yellow, respectively.

The notion of a group concerns much more general entities than numbers. The symmetries (i.e. rotations and reflections) of a square form a group called a dihedral group, and denoted D₄. The following symmetries occur:

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square by 90°, 180°, and 270°, denoted by r₁, r₂ and r₃, respectively;
• reflections about the vertical and horizontal middle line (f_h and f_v), or through the two diagonals (f_d and f_c).

Any two symmetries a and b can be composed, i.e. applied one after another. 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 stems from composition of functions).

For example, rotating by 90° left (r₃) and then flipping horizontally (f_h) is the same as performing a reflection along the diagonal (f_d). Using the above symbols, one has (highlighted in blue in the group table):

f_h • r₃ = f_d.

Given this set of symmetries and the described operation, the group axioms can be understood as follows:

1. The closure axiom demands that the composition b • a of any two symmetries a and b is still a symmetry. Another example for the group operation is

   r₃ • f_h = f_c,

   i.e. rotating left by 270° after flipping horizontally equals flipping along the counter-diagonal (f_c). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
2. The associativity constraint is the natural axiom to impose in order to make composing more than two symmetries well-behaved: given three elements a, b and c of G, there are two possible ways of computing "a after b after c". The requirement

   (a • b) • c = a • (b • c)

   means that composing a after b, and calling this symmetry x, then x after c is the same as a after y, where y in turn is applying b after c. For example, (f_d • f_v) • r₂ = f_d • (f_v • 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.

3. 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.

4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of the identity id, the flips f_h, f_v, f_d, f_c and the 180° rotation r₂ is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r₃ and r₁ are each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols,

   f_h • f_h = id,

   r₃ • r₁ = r₁ • r₃ = id.

Definition

The definition of a group is an abstract formulation incorporating the essential features common to the integers and the above symmetry group: a group (G, •) is a set G with a binary operation • on G that satisfies the following four axioms:

1.	Closure.	For all a, b in G, the result of the operation a • b is also in G.
2.	Associativity.	For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds.
3.	Identity element.	There exists an element e in G, such that for all elements a in G , the equation e • a = a • e = a. holds.
4.	Inverse element.	For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

The order in which the group operation is carried out can be significant. 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 does always hold in the group of integers under addition, because a + b = b + a for any two integers. However, it does not always hold in the symmetry group D₄: f_h • r₁ = f_c but r₁ • f_h = f_d. Groups for which the equation a • b = b • a always holds are known as abelian groups. Thus, the integer addition group is abelian, but the D₄ symmetry group is not.

History

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 of solutions of polynomial equations of degree higher than 4. French mathematician Évariste Galois (1811–1832), extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. Galois' ideas were first rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θⁿ = 1 (1854) gives the first abstract definition of a finite group.

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. The study of Lie groups was founded in 1884 by Sophus Lie.

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. Early attempts to prove Fermat's Last Theorem were led to a climax by Ernst Kummer in 1847, who developed groups describing factorization into prime numbers.

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). Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group. The early 20th century's group theory encompassed roughly the content of the basic concepts (see below). Group theory subsequently grew in both depth and breadth, branching into areas such as algebraic groups, group extensions, and representation theory. A massive collaborative effort, initiated in the 1950s, led to the classification of all finite simple groups in 1982. Research is ongoing to complete and simplify the proof of this classifications.

Simple consequences of the group axioms

Elementary group theory is concerned with basic facts about all groups, which can be obtained directly from the group axioms. For comparison, the study of a particular group, such as the above D₄ symmetry group, can be more involved, e.g., identifying its subgroups, quotient groups or representations.

For example, repeated applications of the associativity axiom show that the unambiguity 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.

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, we suppose that there are two inverses, denoted l and r, and we proceed to show that these two must equal each other

l = l • e = l • (a • r) = (l • a) • r = e • r = r.

Similarly, if the equation n • m = e holds (or m • n = e), that suffices to conclude that n is the inverse element of m.

The inverse of a product is the product of the inverses in the opposite order: (a • b)⁻¹ = b⁻¹ • a⁻¹. To prove this it is enough (by the previous remark, applied to m=a • b and n=b⁻¹ • a⁻¹) to show the identity (a • b) • (b⁻¹ • a⁻¹) = e:

(a • b) • (b−1 • a−1)	=	((a • b) • b−1 ) • a−1	(associativity)
	=	(a • (b • b−1)) • a−1	(associativity)
	=	(a • e) • a−1	(definition of inverse)
	=	a • a−1	(definition of identity element)
	=	e	(definition of inverse).

Division

In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b. In fact, right multiplication of the equation by a⁻¹ gives the solution x = x • a • a⁻¹ = b • a⁻¹. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a⁻¹ • b. In general, x and y need not agree.

Variants of the definition

Group-like structures

	Closure	Associative	Identity	Cancellation	Commutative
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Commutative Groupoid	Unneeded	Required	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutative magma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Commutative quasigroup	Required	Unneeded	Unneeded	Required	Required
Associative quasigroup	Required	Required	Unneeded	Required	Unneeded
Commutative-and-associative quasigroup	Required	Required	Unneeded	Required	Required
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Commutative unital magma	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded
Commutative loop	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Commutative semigroup	Required	Required	Unneeded	Unneeded	Required
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required

Some definitions of a group use seemingly weaker conditions for identity and inverse elements. For instance, the axioms may be weakened to assert only the existence of a left identity and a left inverse for every element. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one above. The closure axiom is already implied by the condition that • be a binary operation. Many authors (such as Lang (2002)) therefore omit this axiom.

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 integers under multiplication (Z, •) are an example of a monoid (see below). 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. The table gives a list of several structures more general than groups.

Notations

Various notations for group operations
	operation	identity	inverse of a
additive notation	+	0	−a
multiplicative notation	*, •, ×	1	a−1
notation related to functions	∘	id, 1	a–1

The notation for groups often depends on the context and the nature of the group operation. There is a tendency to denote abelian groups additively, whereas non-abelian groups are often written multiplicatively. In many situations, there is only one possible (or reasonable) group operation on a given set, therefore it is very common to drop the operation symbol and leave it to the reader to know the context and the group operation. For example the groups (Z_n, +) and (F_q^×, •), the multiplicative group of nonzero elements in the finite field F_q are commonly denoted Z_{n and Fq^×, because only one of the two ring operations makes these sets into a group.}

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.

The arsenal of basic group theory consists of various methods to handle groups. The structure of groups can be understood by breaking them into pieces called subgroups and quotient groups. Groups can be combined into larger groups using direct products and semidirect products. Another important technique, fundamental to group theory, is the comparison of groups via homomorphisms. A particularly well-understood class of groups are the abelian groups. These basic concepts form the standard introduction to groups (see, for example, the books of Lang (2002, 2005) and Herstein (1996, 1975)).

Subgroups

Main article: Subgroup

Informally, a subgroup is a group contained in a bigger one. More precisely, a subset H of G is called a subgroup if the restriction of • to H is a group operation on H. In other words, the identity element of G is contained in H, and whenever g and h are in H, then so are g • h and g⁻¹. 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 rotation in the opposite direction. The subgroup test is a necessary and sufficient condition for a 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 to understand the group as a whole.

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is also the smallest subgroup of G containing S. In the introductory example above, the subgroup generated by r₂ and f_v consists of these two elements, the identity element id and f_h = f_v • 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

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 coset of H containing g are

gH = {gh, h ∈ H} and Hg = {hg, h ∈ H}, respectively.

The cosets of any subgroup H form a partition of G; that is, two left cosets are either equal or have an empty intersection. The same holds of 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. One may then speak simply of the set of cosets of N.

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 = f_vR = {f_v, f_d, f_h, f_c} (highlighted in green). The subgroup R is also normal, because f_vR = U = Rf_v and similarly for any element other than f_v.

Quotient groups

Main article: Quotient group

The quotient group or factor group

G / N = {gN, g ∈ G}, "G modulo N"

treats the cosets of a normal subgroup N as a group. The group operation on this set (sometimes called coset multiplication, or coset addition) behaves in the most natural way possible: (gN) • (hN) = (gh)N for all g and h in G. The coset eN = N itself serves as the identity in this group, and the inverse of Ng in the quotient group is (gN)⁻¹ = (g⁻¹)N.

•	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 = f_vR. The group operation on the quotient is shown at the right. For example, U • U = f_vR • f_vR = (f_v • f_v)R = R.

Quotient 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 = f_v the vertical (or any other) flip), 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 = (rf)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.

Products

Main articles: Direct product and Semidirect product

Taking subgroups and quotients of a given group are methods to break groups into smaller pieces. Building bigger groups from smaller ones can be done, for example, by a technique known as the direct product G×H of two groups G and H. (Here, "product" has a different meaning than the product of elements in a group.) It consists of pairs (g, h), g in G and h in H, with the group operation

(g₁, h₁) • (g₂, h₂) = (g₁ • g₂, h₁ • h₂).

A further generalization of the direct product of two groups is the semidirect product; it allows for the twisting of the group operation on one factor. The group of symmetries of the square (described above) is a semidirect product of the subgroup R consisting of rotations with the corresponding quotient (generated by a reflection).

Group homomorphisms

Main article: Group homomorphism

Group homomorphisms are functions that preserve group structure. The structure being determined by the group operation, this is made formal by requiring

a(g • k) = a(g) • a(k).

for a function a: G → H and any two elements g, k in G. This requirement ensures that a(1_G) = 1_H, and also a(g)⁻¹ = a(g⁻¹) for all g in G, so the additional data from the group axioms are respected, as well.

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 the two possible ways) equal the identity function of G and H, respectively, i.e. 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 practically the same information. For example, proving that g • g = 1 for some element g of G is equivalent to proving that a(g) • a(g) = 1, because applying a to the first equality yields the second, and applying b to the second gives back the first. The category of groups is an abstract framework containing groups and group homomorphisms.

For any group homomorphism a: G → H, the kernel ker a = {g in G : a(g) = 1_H} is the set of elements in G which are mapped to the identity in H. The kernel and image a(G) = {a(g), g ∈ G} of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The First Isomorphism Theorem states that the image of a group homomorphism, a(G) is isomorphic to the quotient group G/ker a.

Abelian groups

Main article: Abelian group

A group G is said to be abelian (in honor of Niels Henrik Abel), or commutative, if the operation satisfies the commutative law

a • b = b • a

for all group elements a and b. If not, the group is called non-abelian or non-commutative. The study of 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.

As noted in the first section, the group of symmetries of the square (discussed above) is non-abelian. However, the subgroup R = {id, r₁, r₂, r₃} consisting of the rotations, as well as the quotient with respect to this subgroup, are abelian. This fact is reflected in the semi-direct product structure of this group (see above).

Cyclic groups

The 6^th complex roots of unity form a cyclic group. ζ is a primitive element, but ζ² is not, because the odd powers of ζ are not a power of ζ².

Main article: Cyclic group

A cyclic group is a group all of whose elements are powers (when the group operation is written multiplicatively) or multiples (written additively) 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. Such an element a is called a generator or a primitive element of the group.

The eponym for this class of groups is the group of n-th complex roots of unity, given by complex numbers ω satisfying ωⁿ = 1 (and whose operation is multiplication). Any cyclic group with n elements is isomorphic to this group. 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.

Examples and applications

A periodic wallpaper gives rise to a wallpaper group.

The fundamental group of a plane minus a point (bold) consists of loops in this area. The blue loop is considered null-homotopic (and thus irrelevant), because it can be shrunk to a point. The presence of the hole prevents the orange loop from being shrunk. This way, the fundamental group detects the hole.

Main articles: Examples of groups and Applications of group theory

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. A major theme in contemporary mathematics is to study given objects by associating groups to them. 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. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.

In addition to the above theoretical applications, many practical applications of groups exist. Cryptography stakes on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are by no means restricted to mathematics. Other sciences such as physics, chemistry or computer science benefit from the abstract group concept, as well.

Numbers

Integers

The integers Z under addition form a group (Z, +), described above. In addition to merely being a group, this group is also abelian because

a + b = b + a (commutativity of addition).

The integers, with the operation of multiplication instead of addition, denoted (Z, ·) do not form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse.

Rationals

The desire for the existence of multiplicative inverses suggests considering fractions

${\frac {a}{b}}$.

Fractions of integers (with b nonzero) are known as rational numbers. 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}, ·). Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division is possible, such as in Q – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.

Cyclic multiplicative groups

In (Q \ {0}, ·), there are cyclic subgroups

G = {aⁿ, n ∈ Z}

where aⁿ is the n-th exponentiations of the primitive element a of that group. For example, if a is 2 then

G = {..., 2⁻², 2⁻¹, 2⁰, 2¹, 2², ...} = {..., 0.25, 0.5, 1, 2, 4, ...}.

This group is an example of a free abelian group of rank one: the rank is one, because G is generated by one element (a or equivalently a⁻¹) and the freeness refers to the fact that no relations between the powers of this generator occur. Therefore, G, is isomorphic to the (additive) group of integers (Z, +) above. This example shows that distinguishing between additive and multiplicative groups is merely a matter of notation.

Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p. Its elements are integers not divisible by p, considered modulo p. The latter means that two numbers are considered equivalent if their difference is divisible by p. For example, if p = 5, then 4 · 4 = 1 in this group, because the usual product 16 is equivalent to 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. 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 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, because it turns out to be the multiplicative group of nonzero elements in the finite field F_p, denoted F_p^×.

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 order of the group G. An important class are the symmetric groups S_N, the groups of permutations of N letters. For example, the symmetric group on 3 letters S₃ is the group consisting of all possible swaps of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S_N for a suitable N (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 $\underbrace {a\cdot \ldots \cdot a} _{n}$, i.e. application of the operation • to n copies of a. (If • represents multiplication, then aⁿ corresponds to the n^th power of a.) If no such n exists 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 (necessarily finite) subgroup H divides the order of G. The Sylow theorems give a partial converse.

The dihedral group (discussed above) is a finite group of order 8. The order of r₁ is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements f_v etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups F_p^× above have order p−1. The latter groups are crucial to public-key cryptography.

Classification of finite simple groups

Given any mathematical notion, mathematicians often strive for a complete classification (or list) of them. In the context of finite groups, this aim quickly leads to very deep mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Z_p. Groups of order p² can also be shown to be abelian, a statement which does not generalize to order p³, as the non-abelian group D₄ of order 8 = 2³ above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan-Hölder theorem exhibits simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. The monstrous moonshine conjectures, proven by 1998 Fields Medal winner Richard Borcherds, provide a surprising and deep connection between the largest finite simple sporadic group, called the monster group, and modular functions and string theory.

Symmetry groups

Main article: Symmetry group

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. Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. This remark is formalized and exploited using the notion of group actions, which means that every group element performs some operation on another mathematical object, in a way compatible to the group structure. This way, the group leaves its footprints on the mathematical object. In the example below, a group element of order 7 acts on the tiling by permuting the highlighted warped triangles (and the other ones, too).

In chemical fields, such as crystallography, space groups, point groups and their character tables are used to describe molecular symmetries. Not only are groups useful to get hold of symmetries of molecules, but surprisingly they sometimes also prevent molecules from being perfectly symmetric, as in the Jahn-Teller effect, an electronic effect phenomenon predicting the distortion of otherwise perfectly symmetric molecules.

Buckminsterfullerene displays icosahedral symmetry.	Ammonia, NH3. Its symmetry group is cyclic of order 3.	Cubane C8H8 features octahedral symmetry.	Hexaaquacopper(II) complex ion, [Cu(OH2)6]2+. Compared to a perfectly symmetrical shape, the molecule is dilated by about 22% in the vertical direction.	The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, or in CD players. Further applications include differential Galois theory, a domain studying which functions have antiderivatives of a prescribed form, which is able to give group-theoretic criteria when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.

The circle of center 0 and radius 1 in the complex plane is a Lie group under complex multiplication. Being a manifold means that every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

Lie groups

Main article: Lie group

In many situations groups are endowed with an additional structure. Lie groups (in honor of Sophus Lie) are groups which also have a compatible manifold structure, i.e. spaces looking locally like some Euclidian space of the appropriate dimension. Because of the manifold structure it is possible to consider continuous paths in the group. For this reason they are also referred to as continuous groups.

Lie groups are of fundamental importance in physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models – imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example 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.

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.

General linear group and matrix groups

Many groups, especially Lie groups, can be described as groups of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Being an open subset of the space of all n-by-n matrices, it is a Lie group. Its subgroups are referred to as matrix 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.

Representation theory

Main article: Representation theory

Representation theory is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its actions on other spaces. A broad class of group representations are linear representations, i.e. the group is acting on a vector space, such as the 3-dimensional Euclidian space R³. A representation of G on an n-dimensional real vector space is simply a group homomorphism

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.

Given a group action, this gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are particularly useful for finite groups, Lie groups, algebraic groups and (locally) compact groups.

Galois groups

Main article: Galois group

Galois groups are groups of substitutions of the solutions of polynomial equations. For example, the solutions of the quadratic equation ax² + bx + c = 0 are given by

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.$

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 (in particular their solvability) associated to polynomials give a criterion which polynomials do have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.

See also

• List of group theory topics
• List of small groups

Notes

^ a: Because a two-sided inverse of a (i.e. an element c such that c•a and a•c both equal e) is guaranteed to exist, the one-sided inverse b must be equal to c.
^ b: In the context of commutative algebra, Z_n is usually denoted Z/nZ or Z/n, see for example Eisenbud (1995).
^ c: However, a group is not determined by its lattice of subgroups. See Suzuki (1951).
^ d: The fact that the group operation extends this canonically is an instance of a universal property.
^ e: For example, if G is finite, then the size of any subgroup and any quotient group divides the size of G, according to Lagrange's theorem.
^ f: The word homomorphism derives from Greek ὁμός – the same and μορφή – structure.
^ g: The kernel of a homomorphism is always a normal subgroup. The converse statement is also true: any normal subgroup N is the kernel of the canonical map G → G/N, see Lang (2005, §II.4, p. 45).
^ h: The additive notation for elements of a cyclic group would be t • a, t in Z.
^ i: See the Seifert–van Kampen theorem for an example.
^ j: An example is group cohomology of a group which equals the singular homology of its classifying space.
^ k: Elements which do have multiplicative inverses are called units, see Lang (2002, §II.1, p. 84).
^ l: The transition from the integers to the rationals by adding fractions is generalized by the quotient field.
^ m: The same is true for any field F instead of Q. See Lang (2005, §III.1, p. 86).
^ n: For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang (2002, Theorem IV.1.9). The notions of torsion of a module and simple algebras are other instances of this principle.
^ o: The stated property is a possible definition of prime numbers. See prime element.
^ p: For example, the Diffie-Hellman protocol uses the discrete logarithm.
^ q: The groups of order at most 2000 are known. Up to isomorphism, there are about 49 billion. See Besche, Eick, and O'Brien (2001).
^ r: The gap between the classification of simple groups and the one of all groups lies in the extension problem, a problem too hard to be solved in general. See Aschbacher (2004, p. 737).
^ s: Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See Michler (2006), Carter (1989).
^ t: More rigorously, every group is the symmetry group of some graph, see Frucht (1939).
^ u: More precisely, the monodromy action on the vector space of solutions of the differential equations is considered. See Kuga (1993, pp. 105–113).
^ v: See Schwarzschild metric for an example where symmetry greatly reduces the complexity of physical systems.
^ w: This was crucial to the classification of finite simple groups, for example. See Aschbacher (2004).
^ x: See, for example, Schur's Lemma for the impact of a group action on simple modules. A more involved example is the Leray spectral sequence relating arithmetic information to geometric information via the action of the (absolute) Galois group.

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.
• Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7, MR2286236.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hall, G. G. (1967), Applied group theory, American Elsevier Publishing Co., Inc., New York, MR0219593, an elementary introduction
• Herstein, Israel Nathan (1996), Abstract algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR1375019.
• Herstein, Israel Nathan (1975), Topics in algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR0356988
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211, Berlin, New York, ISBN 978-0-387-95385-4, MR1878556{{citation}}: CS1 maint: location missing publisher (link)
• Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3
• Ledermann, Walter (1953), Introduction to the theory of finite groups, Oliver and Boyd, Edinburgh and London, MR0054593
• Ledermann, Walter (1973), Introduction to group theory, New York: Barnes and Noble, OCLC 795613
• Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6

Special references

• 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, ISSN 0002-9920.
• Becchi, C. (1997), Introduction to Gauge Theories, retrieved 2008-05-15
• 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, MR1826989
• Bishop, David H. L. (1993), Group theory and chemistry, New York: Dover Publications, ISBN 978-0-486-67355-4
• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR1102012
• Carter, Roger W. (1989), Simple groups of Lie type, New York: John Wiley & Sons, ISBN 978-0-471-50683-6
• Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, ISSN 0138-4821, MR1865535
• Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Berlin, New York: Springer-Verlag, ISBN 978-3-540-52977-4, MR1075994
• Denecke, Klaus; Wismath, Shelly L. (2002), Universal algebra and applications in theoretical computer science, London: CRC Press, ISBN 978-1-58488-254-1
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1; 978-0-387-94269-8, MR1322960 {{citation}}: Check |isbn= value: invalid character (help)
• Fröhlich, Albrecht (1968), Formal groups, Lecture notes in mathematics, vol. 74, Berlin, New York: Springer-Verlag
• Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe", Compositio Mathematica, 6: 239–50, ISSN 0010-437X
• Goldstein, Herbert (1980), Classical Mechanics (2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588–596, ISBN 0-201-02918-9
• Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1
• Jahn, H.; Teller, E. (1937), "Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934-1990), 161 (905): 220–235, doi:10.1098/rspa.1937.0142
• Kassel, Christian (1994), Quantum Groups, Springer, ISBN 978-0387943701
• Kuipers, Jack B. (1999), Quaternions and rotation sequences - A primer with applications to orbits, aerospace, and virtual reality, Princeton University Press, ISBN 978-0-691-05872-6, MR1670862
• Kuga, Michio (1993), Galois' dream: group theory and differential equations, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3688-3, MR1199112
• Kurzweil, Hans; Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-40510-0, MR2014408
• Lay, David (2003), Linear Algebra and Its Applications, Addison-Wesley, ISBN 978-0-201-70970-4
• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
• Michler, Gerhard (2006), Theory of finite simple groups, Cambridge University Press, ISBN 978-0-521-86625-5
• Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR1304906
• Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR2044239
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
• Romanowska, A.B.; Smith, J.D.H. (2002), Modes, World Scientific, ISBN 9789810249427
• Ronan, Mark (2007), Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, Oxford University Press, ISBN 978-0-19-280723-6
• Rosen, Kenneth H. (2000), Elementary number theory and its applications (4th ed.), Addison-Wesley, ISBN 978-0-201-87073-2, MR1739433
• Rudin, Walter (1990), Fourier Analysis on Groups, Wiley Classics, Wiley-Blackwell, ISBN 047152364X
• Seress, Ákos (1997), "An introduction to computational group theory", Notices of the American Mathematical Society, 44 (6): 671–679, ISSN 0002-9920, MR1452069
• Serre, Jean-Pierre (1977), Linear representations of finite groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90190-9, MR0450380
• Suzuki, Michio (1951), "On the lattice of subgroups of finite groups", Transactions of the American Mathematical Society, 70 (2): 345–371, doi:10.2307/1990375
• Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90894-6
• Weinberg, Steven (1972), Gravitation and Cosmology, New York: John Wiley & Sons
• Welsh, Dominic (1989), Codes and cryptography, Oxford: Clarendon Press, ISBN 978-0-19-853287-3
• Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8

Historical references

• Historically important publications in group theory.
• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5
• Galois, Évariste (1908), Tannery, Jules (ed.), Manuscrits de Évariste Galois, Paris: Gauthier-Villars (Galois work was first published by Joseph Liouville in 1843).
• Kleiner, Israel (1986), "The evolution of group theory: a brief survey", Mathematics Magazine, 59 (4): 195–215, ISSN 0025-570X, MR863090
• Smith, David Eugene (1906), History of Modern Mathematics, Mathematical Monographs, No. 1
• Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: Dover Publications, ISBN 978-0-486-45868-7

External links

• Weisstein, Eric W. "Group". MathWorld.
• Group at PlanetMath.
• The development of group theory at The MacTutor History of Mathematics archive.