Group (mathematics)

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The hours on a clock form a group under modular addition.

A group is one of the fundamental objects of study in the field of mathematics known as abstract algebra. The branch of algebra that studies groups is called group theory. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.

Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-) direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics Glossary of group theory
Finite groups Classification of finite simple groups Cyclic group Z/n Symmetric group S_n Dihedral group D_n Alternating group A_n Mathieu groups M₁₁ M₁₂ M₂₂ M₂₃ M₂₄ Conway groups Co₁ Co₂ Co₃ Janko groups J₁ J₂ J₃ J₄ Fischer groups F₂₂ F₂₃ F₂₄ Baby Monster group (B) Monster group (M)
Discrete groups Lattices Integers (Z) Lattice Modular groups PSL(2,Z) SL(2,Z)
Topological / Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G₂ F₄ E₆ E₇ E₈ Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Elliptic curve Linear algebraic group Abelian variety

Many investigated structures in mathematics turn out to be groups. These include familiar number systems, such as: the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers under multiplication. Other important examples are: the group of non-singular matrices under multiplication, and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting.

Definition

A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:

Closure: For all a, b in G, the result of a * b is also in G.

Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).

Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.

Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.

Some texts omit the explicit requirement of closure, since the closure of the group follows from the definition of a binary operation.

Using the identity element property, it can be shown that a group has exactly one identity element. See the proof below.

The inverse of an element a can also be shown to be unique, and it is usually written a^-1 (but see the notation below for additively written groups).

A group (G, *) is often denoted simply G where there is no ambiguity as to what the operation is.

Illustration of definition

The square.

An example will explain some properties of groups. Consider a square. We are interested in the symmetries of the square. There are the following types of symmetries:

rotation about 90°, 180° and 270° (clockwise). We will write these symmetries as rot_90°, rot_180°, and rot_270°, respectively. Note that the counter-clockwise rotations are included here, for example rotating 270° clockwise is equal to rotating 90° counter-clockwise.
reflection along the vertical or horizontal middle line, or along the two diagonals. Let us write the reflections as ref_V, ref_H, ref_D1 and ref_D2, respectively.
Finally, the identical operation id leaving everything unchanged is also a symmetry.

All of them keep the shape of the square unchanged. (In the images, the vertices are colored only for making clear the operations).


Clockwise rotation by 90° rot_90°	Clockwise rotation by 180° rot_180°	Clockwise rotation by 270° rot_270°	Reflection along the vertical ref_V	Reflection along the horizontal ref_H	Reflection along a diagonal ref_D1	Reflection along the other diagonal ref_D2

This set G of symmetries is an example for a group, the so-called dihedral group of order 8. Being a group means the following:

Two symmetries can be composed, i.e. given two symmetries a and b, we can first perform a and then b and the result will still be a symmetry. We write the result b * a (meaning "b after a"). For example, rotating by 270° and then rotating by 180° equals a rotation by 90°, i.e. using the above symbols, we have

rot_180° * rot_270° = rot_90°.

In a more formal language, G is endowed with a binary operation *, i.e. any two elements can be composed to a third element.

Applied to this example group, the definition reads:

Associativity: given three elements a, b and c of G, (a * b) * c = a * (b * c).
Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a. In the example, e is just the symmetry which leaves everything unchanged.
Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element. In the example, rotating a given angle clockwise and then rotating by the same angle counter-clockwise will leave the square unchanged and the same is true if we reverse the order, i.e. first counter-clockwise and then clockwise. Also, reflecting along a diagonal, say, can be inverted by applying the same reflection again. In symbols:

rot_270° * rot_90°=rot_90° * rot_270° = id and ref_D1 *ref_D1 = id.

History

Groups of permutations were already being studied in the 18th century and were applied to solve problems in the theory of equations. However, the formal notion of a group was not published until the late 19th century, and by this time groups had found applications in number theory as well as in geometry.

Basic concepts in group theory

Subgroups

A subset H ⊂ G is called subgroup if the restriction of * to H is a group operation on H. In other words, it is a group using the restriction of the operation defined on G. In the example above, the rotations constitute a subgroup, since a rotation composed with a rotation is still a rotation.

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 g, h ∈ H. The closure, under the group operation and inversion, of any nonempty subset of a group is a subgroup.

If G is a finite group, then so is H. Further, the order of H divides the order of G ( Lagrange's Theorem).

The powers of any element a and their inverses (that is, a⁰ = e, a, a², a³, a⁴, …, a⁻¹, a⁻², a⁻³, a⁻⁴, …) always form a subgroup of the larger group. It is said that a generates that subgroup.

A subgroup H always defines a set of left and right cosets. Given an arbitrary element g in G, the left coset of H containing g is

$gH=\{gh:h\in H\}$

and the right coset of H containing g is

$Hg=\{hg:h\in H\}.$

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. The same holds true of the right cosets of H. In general, the left cosets of H are not necessarily equal to the right cosets of H. If it is the case that for all g in G, gH=Hg, then H is said to be a normal subgroup.

Quotient groups

If N is a normal subgroup of G, its set of left cosets and right cosets are the same and one may speak simply of the set of cosets of N. In this case, 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. Note that 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 $^{-1}$ ).

Simple groups

If a group G is not the trivial group and its only normal subgroups are the trivial group and the group itself, it is called a simple group. With the notion of quotient groups, it can be phrased equivalently as: A group with only the trivial group and the group itself as quotient groups is simple.

Group homomorphisms

If G and H are two groups, a group homomorphism f is a mapping f: G → H that preserves the structure of the groups in question. The structure of groups being determined by the group operation, this means the following: if g and k are any two elements in G, then

f(gk)=f(g)f(k).

This requirement ensures that f(1_G)=1_H, and also f(g)⁻¹=f(g⁻¹) for all g in G.

Two groups G and H are called isomorphic if there exists a group homomorphism f between G and H which is both surjective (onto) and injective (one-to-one).

The kernel of a homomorphism f is denoted ker f and is the set of elements in G which are mapped to the identity in H. That is, ker f={g in G : f(g)=1_H}. The kernel of a homomorphism is always a normal subgroup. The First Isomorphism Theorem states that the image of a group homomorphism, f(G) is isomorphic to the quotient group G/ker f. A useful fact concerning homomorphisms is that they are injective if and only if their kernel is trivial (i.e. ker f={1_G}).

Abelian groups

A group $G$ is said to be abelian, or commutative, if the operation satisfies the commutative law. That is, for all $a$ and $b$ in $G$ , $a*b=b*a$ . If not, the group is called non-abelian or non-commutative. The name "abelian" comes from the Norwegian mathematician Niels Abel. The above example of symmetries of the square is non-abelian, because

rot_90° * ref_V = ref_D2 ≠ ref_D1 = ref_V * rot_90°

The center of a group is a subgroup consisting of the elements which commute with every other element in the group. In a commutative group the center is the whole group; at the other extreme there are groups whose centre is trivial, i.e. it consists only of the identity element.

Cyclic groups

A cyclic group is a group whose elements may be generated by successive composition of the group operation being applied to a single element of that group. An element with this property is called a generator or a primitive element of the group. Cyclic groups are abelian.

A multiplicative cyclic group in which G is the group, and a is a generator:

$G = \{ a^n \mid n \in \Z \}$

An additive cyclic group, with generator a:

$G' = \{ n \cdot a \mid n \in \Z \}$

If successive composition of the operation defining the group is applied to a non-primitive element of the group, a cyclic subgroup is generated. According to Lagrange's theorem, the order of the cyclic subgroup divides the order of the group. Thus, if the order of a finite group is prime, all of its elements, except the identity, are primitive elements of the group.

Order of groups and elements

The order of a group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G. If the order is not finite, then the group is an infinite group, denoted |G| = ∞.

The order of an element a in a group G is the least positive integer n such that aⁿ = e, where aⁿ represents $\underbrace{a * \cdots * a}_n$ , 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.

The order of the above sample group is eight, the order of rot_90° is four, because rotating 4 times by 90° is not changing anything. The order of the reflection elements ref_V etc. is two.

Notations and remarks

Group operation

Groups can use different notation depending on the context and the group operation.

Additive groups use + to denote addition, and the minus sign – to denote inverses. For example, a + (–a) = 0 in Z.
Multiplicative groups use *, $\cdot$ , or the more general 'composition' symbol $\circ$ to denote multiplication, and the superscript ^–1 to denote inverses. For example, a*a^–1 = 1. It is very common to drop the * and just write aa^–1 instead.
Function groups use • to denote function composition, and the superscript ^–1 to denote inverses. For example, g • g^–1 = e. It is very common to drop the • and just write gg^–1 instead.

Omitting a symbol for an operation is generally acceptable, and leaves it to the reader to know the context and the group operation.

When defining groups, it is standard notation to use parentheses in defining the group and its operation. For example, (H, +) denotes the group formed by the set H with addition as group operation. For groups like (Z_n, +) and (F_q^*, *), the multiplicative group of nonzero elements in the finite field F_q, it is common to drop the parentheses and the operation (since only one operation makes these set into a group), as Z_n and F_q^*. It is also correct to refer to a group by its set identifier, e.g. H or $\Z$ , or to define the group in set-builder notation, provided it is clear which group operation is intended.

Identity element

Using the identity element property, it can be shown that a group has exactly one identity element. Therefore one usually speaks of the identity: suppose both e and f are identity elements. Then, because f is a (right) identity element e * f = e, and because e is a (left) identity element e * f = f, whence e = f.

The identity element e is sometimes known as the "neutral element," and is sometimes denoted by some other symbol, depending on the group:

In multiplicative groups, the identity element can be denoted by 1.
In invertible matrix groups, the identity element is usually denoted by I or Id.
In additive groups, the identity element may be denoted by 0.
In function groups, the identity element is usually denoted by f₀.

If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products {xs : s in S}; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets (see cosets).

Inverse

The inverse of an element a can also be shown to be unique, and it is usually written a⁻¹ or −a, depending on the context. Suppose given an inverse l and another inverse r. Then

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

Moreover, if in a group we know only that b * a = e, then this suffices to conclude that b is the inverse element of a (since a two-sided inverse of a is guaranteed to exist, and then b must be equal to it). Similarly a * b = e suffices for the same conclusion.

(However, a set with a binary operation can have many left identity elements or many right identity elements, provided it has none of the opposite kind: take for instance on any set the operation defined by a * b = b, then any element is a left identity element, but none is a right identity element. Similarly in a monoid an element can have multiple left inverse elements, provided it has no right inverse elements (and vice versa): the set of all maps from an infinite set X to itself is a monoid under function composition, in which every injective map has a left inverse, and every surjective map has a right inverse, but neither of these inverses is unique in general. Yet if all elements in a monoid have a left inverse, the monoid can be shown to be a group.)

Associativity

For a sequence of multiple factors in a given order, one can form a product in many different ways by inserting parentheses; however, by several applications of the associativity property, any two of these can be shown to be equal. For this reason the expression

a₁ * a₂ * ··· * a_n

is unambiguous and parentheses are usually omitted in such expressions. As a consequence it is hardly ever necessary to explicitly invoke the associativity property.

Variants of the definition

Some definitions of a group use seemingly weaker conditions for identity and inverse elements. Instead of requiring a two-sided identity element, one may separately require the existence of a left and right identity element, and similarly one may separately require the existence of a left and right inverse elements: in both cases the left and right elements can be shown to be the same (and each is unique).

Examples of groups

The integers under addition

The probably most familiar group is the group of integers under addition. One can think of the axioms of a group being modelled on the properties of the integers Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, together with the group operation "+", which denotes, as usual, the addition. The axioms to be checked are:

Closure: If a and b are integers then a + b is an integer.
Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.

This group is also abelian because a + b = b + a.

If we extend this example further by considering the integers with both addition and multiplication, it forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group.)

Some multiplicative groups

The term multiplicative group refers to groups whose operation stems from multiplication in a certain sense (depending on the context).

The integers under multiplication

To begin with, we give a counterexample: the integers with the operation of multiplication, denoted by "·". According to general notation, this is denoted (Z, ·). 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. We cannot choose b = 1/2 because 1/2 is not an integer. Since not every element of (Z, ·) has an (multiplicative) inverse, (Z, ·) is not a group. It is, however, a commutative monoid, which is a similar structure to a group but does not require inverse elements.

The nonzero rational numbers

The natural step to remedy this is considering the set of rational numbers Q, the set of all fractions of integers a/b, where a and b are integers and b is nonzero, and the multiplication operation, again denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q, ·), like (Z, ·), is not a group.

However, if we instead use the set of all nonzero rational numbers Q \ {0}, then (Q \ {0}, ·) does form an abelian group. Indeed, closure, associativity and identity element axioms are easy to check and follow from the properties of integers (we don't lose closure by 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.

Just as the integers form a ring, the rational numbers form the algebraic structure of a field, allowing the operations of addition, subtraction, multiplication and division.

Cyclic multiplicative groups

In (Q, ·), 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. For example, if a is 2 then

$G = \{ .., 2^{-2}, 2^{-1}, 2^0, 2^1, 2^2, 2^3, .. \} = \{ .., 0.25, 0.5, 1, 2, 4, 8, .. \}.\,$

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 group of integers (under addition) introduced above.

Consindering the group

{aⁿ, n ∈ Z/mZ},

the modulus m binds the group into a finite set with a non-fractional set of elements, since the inverse (and $x^{-2}$ , etc.) would be within the set.

The nonzero integers modulo a prime

The nonzero classes of integers modulo p, a prime number, form a group under multiplication. The product of two integers neither of which is divisible by p is not divisible by p either (because p is prime), which shows that the indicated set of classes is closed under multiplication. Associativity is clear, and the class of 1 is the identity for multiplication, so it remains to prove is that each element has an inverse: given an integer a not divisible by p, one has to find an integer b such that

$a \cdot b \equiv 1 \pmod p$ .

This can be shown by using the Euclid algorithm, for example. Actually, this example is similar to (Q\{0}, ·) above, because it turns out to be the group of nonzero elements in the finite field F_p. However, it is distinctly different from the second multiplicative cyclic group mentioned above.

Finite groups

If the number of elements of a group G is finite, then G itself is called a finite group. The above dihedral group of order 8 is an example. Two important classes are the following:

the cyclic (abelian) groups Z/nZ treated above. Any abelian finite group is a finite direct sum of groups of this kind, this is part of the fundamental theorem of finitely generated abelian groups.
the symmetric group S_N: it is the group 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. Parallel to the group of symmetries of the square above, S₃ can also be interpreted as the group of symmetries of an equilateral triangle.

Cayley's theorem states that any finite (not necessarily abelian) group can be expressed as a subgroup of a symmetric group S_N.

Elementary group theory

Elementary group theory is concerned with basic facts that hold for all individual groups. For example:

You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b. In fact, right respectively left multiplication of the equation by a^-1 gives the solution x = b * a^-1 respectively y = a^-1 * b.

(Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a * b)⁻¹ = b⁻¹ * a⁻¹.

Proof: We will demonstrate that (a * b) * (b^-1 * a^-1) = e, which as mentioned above suffices to prove that b^-1 * a^-1 is the inverse of a * b.

$(ab)(b^{-1}*a^{-1})$	=	$(a(bb^{-1}))*a^{-1}$	(associativity)
	=	$(ae)a^{-1}$	(definition of inverse)
	=	$a*a^{-1}$	(definition of neutral element)
	=	$e$	(definition of inverse)

Constructing new groups from given ones

Besides subgroups and quotient groups are two basic ways of constructing new groups from given ones. Other manipulation techniques include:

Direct product: If (G, *) and (H, •) are groups, then the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂) is a group. The direct product can also be defined with any number of terms, finite or infinite, by using the Cartesian product and defining the operation coordinate-wise.
Semidirect product: If N and H are groups and φ : H → Aut(N) is a group homomorphism, then the semidirect product of N and H with respect to φ is the group (N × H, *), with * defined as

(n₁, h₁) * (n₂, h₂) = (n₁ φ(h₁) (n₂), h₁ h₂)
Direct external sum: The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non-identity coordinates. If the family is finite the direct sum and the product are equivalent.

Generalizations

	Totality*	Associativity	Identity	Inverses	Commutativity
Group-like structures
Magma	Yes	No	No	No	No
Semigroup	Yes	Yes	No	No	No
Monoid	Yes	Yes	Yes	No	No
Group	Yes	Yes	Yes	Yes	No
Abelian Group	Yes	Yes	Yes	Yes	Yes
Loop	Yes	No	Yes	Yes	No
Quasigroup	Yes	No	No	Yes	No
Groupoid	No	Yes	Yes	Yes	No
Category	No	Yes	Yes	No	No
Semicategory	No	Yes	No	No	No
	* Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.

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

Eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid.
A monoid without an identity is called a semigroup.
Alternatively, relaxing the requirement that the operation be associative while still requiring the possibility of division, the resulting algebraic structure is a loop.
A loop without an identity is called a quasigroup.
Finally, dropping all axioms for the binary relation, the resulting algebraic structure is called a magma.

Groupoids, which are similar to groups except that the composition a * b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures. Groupoids, in turn, are special sorts of categories.

Supergroups and Hopf algebras are other generalizations, and so are heaps.

Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond.

Formal group laws are certain formal power series which have properties much like a group operation.

In differential geometry, algebraic geometry, and topology, the group concept specializes to include groups with additional structure. Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets.

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