Group (mathematics)

In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below. For example, the set of integers is a group when addition is taken as the binary operation. The branch of mathematics which studies groups is called group theory.

Group theory originated with the work of Évariste Galois in 1830, which concerned the problem of when an algebraic equation is soluble by radicals. Prior to this work, groups were mainly studied in terms of permutations. Some aspects of abelian group theory were also known in the theory of quadratic forms.

Many of the structures investigated 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.

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.

History

See Group theory.

Basic definitions

A group (G, * ) is a set G together with a binary operation * : G × G → G, satisfying the group axioms below. "a * b" represents the result of applying the operation * to the ordered pair (a, b) of elements of G. The group axioms are the following:

Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
Neutral element: There is an element e in G such that for all a in G, e * a = a * e = a.
Inverse element: For all a in G, there is an element b in G such that a * b = b * a = e, where e is the neutral element from the previous axiom.

You will often also see the axiom:

Closure: For all a and b in G, a * b belongs to G.

The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure. When determining if * is a group operation, however, it is nonetheless necessary to verify that * satisfies closure; this is part of verifying that it is in fact a binary operation.

The neutral element is usually called the identity element for a multiplicative group and the null element or zero element for an additive group.

It can be shown that if there is both a left and a right neutral element, they must be equal (here both e) and there can be only one. This is why we can refer to the neutral element in the third axiom of the definition of a group, even though the second axiom only postulates that the set of the neutral elements is nonempty.

It is worth noting that we can weaken these axioms -- associativity, right identity (i.e., a * e = a) and right inverse (a * b = e) imply the left identity and left inverse as given above. It is canonical to define the axioms as above because combinations of the above define other useful algebraic structures -- e.g., the groupoid and semigroup. Thus the above axioms are not strictly minimal from a logical viewpoint; however, the difference is slight and in practice one usually just checks the above axioms.

It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a * b ≠ b * a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a * b = b * a. Groups lacking this property are called non-abelian.

The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set.

Note that we often refer to the group (G, * ) as simply "G", leaving the operation * unmentioned. But to be perfectly precise, different operations on the same set define different groups.

Notation for groups

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:

We write "a · b" or even "ab" for a * b and call it the product of a and b;
We write "1" (or "e") for the identity element and call it the unit element;
We write "a⁻¹" for the inverse of a and call it the reciprocal of a.

However, sometimes the group operation is thought of as analogous to addition and written additively:

We write "a + b" for a * b and call it the sum of a and b;
We write "0" for the identity element and call it the zero element;
We write "−a" for the inverse of a and call it the opposite of a.

Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a⁻¹ for the inverse of a.

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.

Some elementary examples and nonexamples

An abelian group: the integers under addition

A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively).

Proof:

If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)
If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element)
If a is an integer, then there is an integer b := −a, such that a + b = b + a = 0. (Inverse element)

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

The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

Not a group: the integers under multiplication

On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:

If a and b are integers then a · b is an integer. (Closure)
If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)
1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element)
However, 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. (Inverse element fails)

Since not every element of (Z,·) has an inverse, (Z,·) is not a group. The most we can say is that it is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, that is the set of numbers a/b such that a and b are integers and b is nonzero, and the operation multiplication, 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 Q \ {0} instead of Q, that is include every rational number except zero, then (Q \ {0},·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero.

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. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block".

In multiplicative form, we traditionally write x'y for the combined action "first do y, then do x"; so that a'b is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:

e : RGB → RGB
a : RGB → GRB
b : RGB → RBG
ab : RGB → BRG
ba : RGB → GBR
a'b'a : RGB → BGR

Note that the action a'a has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write a'a = e. Similarly,

bb = e,
(a'b'a)(a'b'a) = e, and
(a'b)(b'a) = (b'a)(a'b) = e;

so each of the above actions has an inverse.

By inspection, we can also determine associativity and closure; note for example that

(a'b)a = a(ba) = a'ba, and
(b'a)b = b(a'b) = b'a'b.

This group is called the symmetric group on 3 letters, or S₃. It has order 6 (or 3 factorial), and is non-abelian (since, for example, a'b ≠ b'a). Since S₃ is built up from the basic actions a and b, we say that the set {a,b} generates it.

Every group can be expressed in terms of permutation groups like S₃; this result is Cayley's theorem and is studied as part of the subject of group actions.

Further examples

For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.

Simple theorems

A group has exactly one identity element.

Every element has exactly one inverse.

Proof: Suppose both b and c are inverses of x. Then, by the definition of an inverse, xb = bx = e and xc = cx = e. But then:

{| xb = e = xc

xb = xc

bxb = bxc

(multiplying on the left by b)

eb = ec

(using bx = e)

b = c

(neutral element axiom)

Therefore the inverse is unique.

The first two properties actually follow from associative binary operations defined on a set. Given a binary operation on a set, there is at most one identity and at most one inverse for any element.

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.

The expression "a₁ * a₂ * ··· * a_n" is unambiguous, because the result will be the same no matter where we place parentheses.

(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 (ab)(b^-1a^-1) = (b^-1a^-1)(ab) = e, as required by the definition of an inverse.

(ab)(b^{-1}a^{-1})

a(bb^{-1})a^{-1}

(associativity) =

aea^{-1}

(definition of inverse) =

aa^{-1}

(definition of neutral element) =

e

(definition of inverse)

And similarly for the other direction.

These and other basic facts that hold for all individual groups form the field of elementary group theory.

Constructing new groups from given ones

If a subset H of a group (G,*) together with the operation * restricted on H is itself a group, then it is called a subgroup of (G,*).
The direct product of two groups (G,*) and (H,•) is the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂). 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.
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 of course the same.
Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.

Generalizations

In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article.

If we eliminate the requirement that every element have an inverse, then we get a monoid.
If we additionally do not require an identity either, then we get a semigroup.
Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop.
If we additionally do not require an identity, then we get a quasigroup.
If we don't require any axioms of the binary operation at all, then we get 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. They are special sorts of categories.

Supergroups and Hopf algebras are other generalizations.

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.

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.

References

Herstein, I.N. Abstract Algebra, Wiley, ISBN 0471368792

Dummit, David and Richard Foote. Abstract Algebra, Wiley, ISBN 0471433349

Lang, Serge. Algebra, Springer, ISBN 038795385X

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