Group (math) encyclopedia topics

Definition and illustration

A group is a set equipped with an operation "•" which assigns to any two elements a and b another element denoted . To build a group, the operation is subject to certain requirements called group axioms. Before giving the formal definition of a group by specifying the four group axioms, the concept is illustrated with two key examples.

First example: the integers

The first example of a group is the set of integers Z containing the numbers

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

Together with the usual addition operation "+" they form what is probably the most familiar group. The group axioms can be thought of as being modeled on the properties of the integers together with the addition operation. The following properties of numbers serve as a model for the abstract group axioms below.

For any two integers a, b, the sum a + b is also an integer. In other words, adding two numbers does not leave the domain Z, a property called closure.
For all integers a, b and c, (a + b) + c = a + (b + c). This says that the two ways of calculating give the same result and is referred to as the associativity of the addition.
If a is any integer, then 0 + a = a + 0 = a. Therefore adding 0 amounts to no change, which is why 0 is named identity element.
For each a in Z, the element b = −a is an integer and satisfies 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°)
f_v (vertical flip)	f_h (horizontal flip)
f_d (diagonal flip)	f_c (counter-diagonal flip)
Elements of the symmetry group. The vertices are colored only to visualize the operations.

group table
•	id	r₁	r₂	r₃	f_v	f_h	f_d	f_c
id	id	r₁	r₂	r₃	f_v	f_h	f_d	f_c
r₁	r₁	r₂	r₃	id	f_c	f_d	f_v	f_h
r₂	r₂	r₃	id	r₁	f_h	f_v	f_c	f_d
r₃	r₃	id	r₁	r₂	f_d	f_c	f_h	f_v
f_v	f_v	f_d	f_h	f_c	id	r₂	r₁	r₃
f_h	f_h	f_c	f_v	f_d	r₂	id	r₃	r₁
f_d	f_d	f_h	f_c	f_v	r₃	r₁	id	r₂
f_c	f_c	f_v	f_d	f_h	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. 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").

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:

The closure axiom demands that the composition b • a of any two symmetries a and b is still a symmetry. Another example is
r₃ • f_h = f_c,
i.e. rotating left by 270° after flipping horizontally equals flipping along the counter-diagonal. Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
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
(f_d • f_v) • r₂ = r₃ • r₂ = r₁, which equals
f_d • (f_v • r₂) = f_d • f_h = r₁.
The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form,
id • a = a,

a • id = a.
An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of the 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.

Formal definition

The formal group definition 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 a • b is also in G.
2.	Associativity.	For all a, b and c in G, the equality (a • b) • c = a • (b • c) holds.
3.	Identity element.	There exists a (unique) element e in G such that for all a in G, one has 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.

The identity a • b = b • a is not required to hold. The equality does hold in the first example of integers with addition, because a + b = b + a for any two integers, but does not hold in the symmetry group example, because f_h • r₁ = f_c, but r₁ • f_h = f_d and similar other examples.

History

Historically, the group concept has evolved in several parallel threads. 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 expressing symmetry features of the solutions of such equations, thus giving a criterion for the solvability of the equation in question. These groups can be viewed as consisting of certain permutations of these solutions; they are nowadays called Galois groups. Permutation groups, which are similar to this concept were investigated in particular by Cauchy. Cayley's On the theory of groups, as depending on the symbolic equation θⁿ = 1 (1854) gives the first abstract definition of finite groups.

Secondly, the systematic use of groups in geometry, mainly in the guise of symmetry groups, was initiated by Klein's 1872 Erlangen program. The study of what are now called Lie 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. 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. Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify all finite simple groups in 1982. Completing and simplifying the proof of the classification are areas of active research.

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. These facts are usually direct consequences of the group definition — obtained by invoking the axioms a few times.

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.

Uniqueness of identity element and inverses

Though the uniqueness of the identity and inverse elements 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.

The following proof of the latter fact shows the flavor of elementary group theory: 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 n • m = e (or m • n = e) 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

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.

Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on G. Many authors (such as ) therefore omit this axiom.

In abstract algebra, more general structures arise by relaxing some of the axioms defining a group, listed in the table. 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.

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. 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 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.}

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 means a function assigning to every element of X an element of Y.

The arsenal of basic group theory comprises various methods to handle 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 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 (see, for example, the books of and ).

Subgroups

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 is and g⁻¹. In the example above, the identity and the rotations constitute a subgroup R = {id, r₁, r₂, and 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 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

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 the elements of G; that is, two left cosets are either equal or have an empty intersection. 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.

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 other group element instead of f_v.

Quotient groups

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 nicest 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 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 graphical device showing certain features of discrete groups.

Products

Taking subgroups and quotients of a given group G tends to reduce the size of G. 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).

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

Group homomorphisms are functions that respect 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 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

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

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

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 wished-for existence of a 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 the rationals with multiplication, being a group: since 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 {{nowrap|(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 .

The inverse b can be found by using that the greatest common divisor equals 1. For example, the inverse of 4 in this group 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

A group is called finite if it has finitely many 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

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 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 at the right, a group element of order 7 acts on the tiling by permuting the highlighted warped triangles (and the other ones, too).

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.

Lie groups

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.

Various Lie groups are important 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. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

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. In chemical fields, such as crystallography, space groups and their character tables are used to describe molecular symmetries.

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

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.

Notes

Citations

References

General references

, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.
, Chapter 5 provides a layman-accessible explanation of groups.
.
, an elementary introduction
.