Group (mathematics)

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Groups

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product direct sum Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko group J1..4 Fischer group F22..24 Baby Monster group B Monster group M Continuous groups Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) E6 E7 E8 F4 G2 SL2 Lorentz group Poincaré group Infinite dimensional groups conformal group diffeomorphism group Quantum group O(∞) SU(∞) Sp(∞)

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A group is one of the fundamental objects of study in the field of mathematics known as abstract algebra, and more specifically group theory.^[1][2] Groups are sets with an additional operation which combines any two elements to a third one, subject to certain conditions familiar from number systems such as the integers, or the rational numbers with addition as the group operation, as well as the non-zero rational numbers with multiplication.

Groups often occur in the guise of symmetry groups of geometrical objects. Groups, in particular Lie groups such as groups of matrices, are essential abstractions in branches of physics involving symmetry principles. Their ability to represent geometric transformations also finds applications in chemistry. Groups are heavily influential in other mathematical domains. Important algebraic structures such as rings and fields can be defined concisely in terms of groups.

Historically, groups are rooted in several parallel origins such as permutation groups, culminating in a notion that allows to investigate properties of groups in a general and abstract setting. Beyond direct implications of the group axioms, basic techniques include studying groups related to a given one (such as sub- or quotient groups) or decomposing groups into simpler parts. A particularly ample theory has been developed for finite groups, coming to the climax of the classification of finite simple groups.

[edit] Definition and illustration

A group (G, •) is a set G with a binary operation • on G that satisfies the following four axioms:^[3]

1.	Closure.	For all a, b in G, the result of a • b is also in G.
2.	Associativity.	For all a, b and c in G, (a • b) • c = a • (b • c).
3.	Identity element.	There exists an element e in G such that for all a in G, e • a = a • e = a.
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.

[edit] First example: the integers

The first example of a group is the set of integers Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}. Under the usual addition operation "+" they form what is probably the most familiar group. It is denoted by (Z, +). The group axioms can be thought of as being modeled on the properties of the integers together with the addition operation. The abstract group axioms reduce to statements about numbers, in this case:

1.	Closure.	For any two integers a, b, the sum a + b is also an integer.
2.	Associativity.	For all integers a, b and c, (a + b) + c = a + (b + c).
3.	Identity element.	If a is any integer, then 0 + a = a + 0 = a. Thus 0 is the (additive) identity.
4.	Inverse element.	For each a in Z, b = −a is an integer and satisfies a + b = b + a = 0. Thus −a is the (additive) inverse of the integer a.

[edit] 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 fd fc 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).

Group theory and the notion of a group concern much more general entities than numbers. The following illustrates the meaning of the group axioms for the dihedral group of symmetries of the square.^[4] The elements of the group are operations which keep the shape of the square unchanged. The operations are:

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

In this example group, the axioms can be understood as follows:

1. The closure axiom demands that any two symmetries can be composed. This is indeed the case—for any two symmetries a and b, we can first perform a and then b and the result will still be a symmetry, written symbolically from right to left ("perform the symmetry b after performing the symmetry a") as:

   b • a.

   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, we have (highlighted in blue in the group table):

   f_h • r₃ = f_d.

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, we check (f_d • f_v) • r₂ = f_d • (f_v • r₂) using the group table at the right:

   (f_d • f_v) • r₂ = r₃ • r₂ = r₁, which equals

   f_d • (f_v • r₂) = f_d • f_h = 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, and

   a • id = a.

4. An inverse element undoes the operation of some other element. In the symmetry group example, 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. Each of the 90° rotations r₃ and r₁ is each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols, for example:

   f_h • f_h = id,

   r₃ • r₁ = r₁ • r₃ = id.

[edit] History

Main article: History of group theory

Historically, the group concept has evolved in several parallel threads.^[5][6][7] One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. Galois, extending prior work of Ruffini and Lagrange, introduced groups of solutions of such equations, thus giving a criterion for the solvability of such equations.^[8] Cauchy pushed the theory of permutation groups further. Cayley's On the theory of groups, as depending on the symbolic equation θⁿ = 1 gives the first abstract definition of finite groups.

Secondly, the relation of groups to geometry was initiated by Klein's 1872 Erlangen program. The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie.

The third root of group theory was number theory: certain abelian group structures had been implicitly used in number-theoretical work by Gauss, and more explicitly by Kronecker.^[9] Early attempts to prove Fermat's last theorem were led to a climax by Kummer by introducing groups describing factorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started with Jordan's Traité des substitutions et des équations algébriques (1870) and von Dyck (1882) who first defined a group in the full abstract sense of this article. The early 20th century's group theory encompassed roughly the content of the basic concepts (see below). Group theory subsequently grew both in depth and in breadth, branching out into areas such as algebraic groups, group extensions and representation theory. The latter was crucial for the success of the classification of finite simple groups in 1982, a major accomplishment of contemporary group theory.^[10]

[edit] First consequences of the group axioms

Elementary group theory is concerned with basic facts about general groups, as opposed for example to the more involved study of groups via their representations.^[11] These facts are usually direct consequences of the group definition — obtained by invoking the axioms a few times — and are often used in group theory without explicit reference to the corresponding statement.^[12]

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. Therefore parentheses are usually omitted in such expressions.

[edit] Uniqueness of identity element and inverses

Though the uniqueness of the identity is not required by the group axioms, it is a consequence of them. Therefore it is customary to speak of the identity, and the inverse of a.^[13]

The following proof of this fact shows the flavor of elementary group theory: suppose both e and f are identity elements. Then

e = e • f = f,

because e is a (left) identity element and f is a (right) identity element. Hence the two identities necessarily agree. Similarly, suppose given two inverses l and r of a fixed element a. Then

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

Moreover, in a group, knowing only that b • a = e (or a • b = e) suffices to conclude that b is the inverse element of a.^[14]

The inverse of a product is the product of the inverses in the opposite order: (a • b)⁻¹ = b⁻¹ • a⁻¹. The identity (a • b) • (b⁻¹ • a⁻¹) = e then suffices to prove that b⁻¹ • a⁻¹ is the inverse of a • b.

(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)

[edit] 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.

[edit] Variants of the definition

Group-like structures
	Totality	Associativity	Identity	Division
Group	Yes	Yes	Yes	Yes
Monoid	Yes	Yes	Yes	No
Semigroup	Yes	Yes	No	No
Loop	Yes	No	Yes	Yes
Quasigroup	Yes	No	No	Yes
Magma	Yes	No	No	No
Groupoid	No	Yes	Yes	Yes
Category	No	Yes	Yes	No

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.^[15]

Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on G. Many authors therefore omit this axiom.^[16]

In abstract algebra, more general structures arise by relaxing some of the axioms defining a group, shown in the table.^[17][18][19] For example, eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid. The integers under multiplication (Z, •) are an example (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 involved kinds of symmetries, often in topological and analytical structures, e.g. the fundamental groupoid.

[edit] Notations

Customary 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.^[20] 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 F_q*, since only one of the two ring operations makes these sets into a group.^[21]

[edit] Basic concepts

Basic notions in group theory
category of groups
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

The arsenal of basic group theory comprises various methods to manipulate groups. The structure of groups can be understood by breaking them into pieces called subgroups and quotient groups. Combining them into larger groups yields direct and semidirect products. An equally important technique, fundamental to the orientation of group theory, is comparing groups using homomorphisms. A particularly well-understood class of groups are the abelian groups. These basic concepts form the standard introduction to groups.^[22]

[edit] 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.^[23] In other words, the identity element of G is contained in H, and whenever g and h are in H, then so is g • h and g⁻¹. In the example above, the rotations {id, r₁, r₂, and r₃} constitute a subgroup (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. It can be read off the group table above, and is indeed a general principle that knowing the subgroups of a group is important to understand the structure of the group in question. 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.

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 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 subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, the same elements) yields an element of this subgroup.

A subgroup H defines a set of 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.^[24]

The set of left cosets of H forms a partition of the elements of G; that is, two left cosets are either equal or have an empty intersection.^[25] The same holds true of the right cosets of H. 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.

[edit] Quotient groups

Main article: Quotient group

Quotient groups, also known as factor groups, treat the cosets of a normal subgroup as a group.^[26] The set of cosets of N may be equipped with an operation (sometimes called coset multiplication, or coset addition) to form a new group, called the quotient group G/N. The operation between the cosets behaves in the nicest way possible: (Ng) • (Nh) = N(gh) for all g and h in G. The coset N itself serves as the identity in this group, and the inverse of Ng in the quotient group is (Ng)⁻¹ = N(g⁻¹).^[27]

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 introductory dihedral group, for example, is presented by two generators r and f (for example, r = r₁, the right rotation and f = f_v the vertical (or any other) flip), together with the relations

r ⁴ = f ² = (rf)² = 1.^[28]

A presentation of a group can also be used to construct the Cayley graph, a graphical device showing certain features of discrete groups.

Taking subgroups and quotients of a given group G tends to reduce the size of G.^[29] Several group constructions reverse this direction, i.e. given two groups, one constructs bigger groups, such as the direct product G×H of the two. (Here, "product" has a slightly 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). ^[30]

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 Z₄ (the subgroup consisting of rotations) with Z₂ (generated by a reflection).

[edit] Group homomorphisms

Main article: Group homomorphism

Group homomorphisms (from Greek μορφη–structure) are mappings that preserve the structure of the groups in question. The structure being determined by the group operation, this is made formal by requiring

a(g • k) = a(g) • a(k).

for a map 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.^[31]

Two groups G and H are called isomorphic if there exist group homomorphisms a between G and H and b: H → G, such that applying the two maps 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. Two isomorphic groups as above 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 the applying a to the first equality yields the second, and applying b to the second gives back the first. This method of dissociating the group from its concrete nature, and focussing instead on the abstract properties is a steadily recurring and deeply impacting theme in algebra, and many other mathematical domains, as well. 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.^[32]

[edit] 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.^[33]

for all group elements a and b. If not, the group is called non-abelian or non-commutative. As the realm of abelian groups is particularly well-understood, many group-related notions, such as the center, or commutators describe the extent to which a given group is not abelian.^[34]

The group of symmetries of the square (discussed above) is non-abelian, because r₁ • f_v = f_c, which is not equal to f_v • r₁ = f_d. The subgroup {id, r₁, r₂, r₃} consisting of the rotations, as well as the quotient with respect to this subgroup, however, are abelian. This fact is reflected in the semi-direct product structure of this group (see above).

[edit] Cyclic groups

Main article: Cyclic group

Cyclic groups are groups whose elements may be generated by successive compositions of the group operation applied to a single element a of that group.^[35] In multiplicative notation, the group therefore consists of the powers

..., 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.^[36] Such an element a is called a generator or a primitive element of the group.

Any cyclic group is abelian. It may or may not be finite. If so, the group is isomorphic to Z_n, where n is the smallest integer such that n • a = 0.^[21] The eponym is the group of n-th complex roots of unity, given by complex numbers ω satisfying ωⁿ = 1.^[37] An infinite cyclic group is isomorphic to (Z, +).^[38]

In any group G, the powers of any group element a and their inverses form a subgroup of G, called the cyclic subgroup generated by a.

[edit] Examples and applications

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, Poincaré founded what is now called algebraic topology by introducing the fundamental and higher homotopy groups. By means of this connection, topological properties translate into group-theoretic properties.^[39] In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.^[40] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include number theory and algebraic geometry. Applications of group theory are by no means restricted to mathematics. Other sciences such as physics, chemistry or computer science benefit from the abstract concept of a group, as well.

[edit] Numbers

[edit] Integers

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

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

The integers are the basic building block for abelian groups, for example every torsion-free group contains (Z, +) as a subgroup.^[41]

The integers, with the operation of multiplication instead of addition, denoted (Z, •) do not form a group. It satisfies the closure, associativity and identity axioms, but fails to have inverses: it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is an integer, but the only solution to the equation ab = 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, so (Z, •) is not a group.^[42]

[edit] Rationals

The wished-for existence of a multiplicative inverses suggests considering fractions

.

Such fractions (with integers a and b and b nonzero) are called 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: since the rational number 0 does not have a multiplicative inverse, (Q, •) is still not a group.

However, the set of all nonzero rational numbers Q \ {0} does form an abelian group under multiplication, denoted (Q \ {0}, •).^[43] 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. Taking addition and multiplication operations together 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.^[44]

[edit] Cyclic multiplicative groups

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

G = {aⁿ, n ∈ Z} ⊂ Q

where aⁿ is the n-th exponentiations of the primitive element a of that group.^[45] 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 – group theory treats groups from a purely abstract point of view, forgetting about the concrete nature of the group elements and the group operation.

[edit] Nonzero integers modulo a prime

For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p.^[46] Its elements are integers not divisible by p, considered modulo p. This means that two numbers are considered equivalent if they give the same remainder when divided by p. For example, if p=5, then 4·3=2 in this group, because the usual product 12 is equivalent to 2, for 12 gives rest 2 when divided by 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. 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 F_p.

[edit] Finite groups

Main article: Finite group

A group is called finite if it has finitely many elements. The number of elements is called order of the group G.^[47] An important class are 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 , i.e. application of the operation • to n copies of the value a. (If • represents multiplication, then aⁿ corresponds to the n^th power of a.) If no such n exists, then the order of a is said to be infinity. The order of an element is the same as 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, by asserting the existence of subgroups whose order is any prime power dividing the order of the group, the p-subgroups.

The dihedral group (discussed above) is a finite group of order 8. The order of r₁ is 4, because rotating four times by 90° does not change anything. The order of the reflection elements f_v etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem.

Given the notion of finite groups, the obvious aim arises to classify (or list) them. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Z/pZ. Groups of order p² can also be shown to be abelian. 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.^[48] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.^[49] The Jordan-Hölder theorem exhibits simple groups as the building blocks for all finite groups.^[50] Listing all finite simple groups was a major achievement in contemporary group theory. Filling the gaps in the 1982 proof and simplifying it are areas of active research.^[51] 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.^[52] From the point of view of applications, using modular arithmetic, finite groups are crucial to public-key cryptography. ^[53]

[edit] Symmetry groups

Main article: Symmetry group

Symmetry groups are groups consisting of symmetries of given mathematical objects – be they of geometric nature, such as introductory symmetry group of the square, or of algebraic nature.^[54] Conceptually, group theory can be thought of as the study of symmetry.^[55] Symmetries greatly simplify the study of geometrical or analytical objects. This remark is formalized and exploited using the notion of group actions. For example, using topological methods such as the monodromy action on the vector space of solutions of certain differential equations, differential Galois theory is able to give group-theoretic criteria when solutions of the equation in question are well-behaved.^[56] Group actions are also much used in geometric invariant theory.^[57] 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.^[58]

[edit] Lie groups

Main article: Lie group

In many situations groups are endowed with an additional structure. The most famous examples are Lie groups (in honor of Sophus Lie). They are groups which also have a (compatible) manifold structure, i.e. spaces looking locally like some euclidian space of the appropriate dimension.^[59] 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.

Various Lie groups are important tools in physics. 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. Lie groups are also of more fundamental importance: Noether's theorem links continuous symmetries to conserved quantities. The Poincaré group plays a pivotal role in special relativity and, by implication, for quantum field theories.^[60] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.^[61]

[edit] General linear group and matrix groups

Most of Lie groups important in physics may be described as groups of matrices together with matrix multiplication: the general linear group GL(n, K) consists of all invertible n-by-n matrices with entries in a fixed field K, for example the real or complex numbers.^[62][63]

The subgroups of GL(n, K) 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.^[64] In chemical fields, such as crystallography, space groups and their character tables are used to describe molecular symmetries.^[65]

[edit] Representation theory

Main article: Representation theory

Representation theory is both an application of the group concept and a major branch of group theory itself.^[66][67] 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, which can be thought of as generalizations of Euclidian space R³. A representation of G on an n-dimensional complex vector space is simply a group homomorphism

ρ: G → GL(n, C)

from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices which is accessible to explicit computations.

Given a group action, this gives further means to study the object being acted on.^[68] 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.

[edit] Galois groups

Main article: Galois group

Galois groups are groups of substitutions of the solutions of polynomial equations.^[69][70] For example, the solutions of the quadratic equation ax² + bx + c = 0 are given by

Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very easy) 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.^[71]

[edit] See also

[edit] Notes

[edit] References

[edit] 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, MR2286236, ISBN 978-0-471-43334-7 .
• Fulton, William & Harris, Joe (1991), Representation Theory, A First Course, vol. 129, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97495-8 
• 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., MR1375019, ISBN 978-0-13-374562-7 .
• Herstein, Israel Nathan (1975), Topics in algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR0356988 
• Lang, Serge (2002), Algebra, vol. 211, Graduate Texts in Mathematics, Berlin, New York, MR1878556, ISBN 978-0-387-95385-4 
• 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 
• Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6 

[edit] 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”, Notices of the American Mathematical Society 51 (7): 736–740, ISSN 0002-9920, <http://www.ams.org/notices/200407/fea-aschbacher.pdf> .
• Borel, Armand (1991), Linear algebraic groups, vol. 126 (2nd ed.), Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 
• 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, MR1865535, ISSN 0138-4821, <http://arxiv.org/abs/math.MG/9911185> 
• Eisenbud, David (1995), Commutative algebra, vol. 150, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8 
• Fröhlich, Albrecht (1968), Formal groups, vol. 74, Lecture notes in mathematics, Berlin, New York: Springer-Verlag 
• Kassel, Christian (1994), Quantum Groups, Springer, ISBN 978-0387943701 
• Kurzweil, Hans & Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, MR2014408, ISBN 978-0-387-40510-0 
• 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.), Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Berlin, New York: Springer-Verlag, MR1304906, ISBN 978-3-540-56963-3 
• 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 
• Serre, Jean-Pierre (1977), Linear representations of finite groups, Berlin, New York: Springer-Verlag, MR0450380, ISBN 978-0-387-90190-9 
• Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8 

[edit] Historical references

• Historically important publications in group theory.
• Kleiner, Israel (1986), “The evolution of group theory: a brief survey”, Mathematics Magazine 59 (4): 195–215, MR863090, ISSN 0025-570X, <http://www.jstor.org/sici?sici=0025-570X(198610)59%3A4%3C195%3ATEOGTA%3E2.0.CO%3B2-9> 
• Smith, David Eugene (1906), History of Modern Mathematics, Mathematical Monographs, No. 1, <http://www.gutenberg.org/etext/8746> 
• 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 

[edit] External links

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