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Group (mathematics)

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

The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms.

A great many of the objects 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, each under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right.

Groups underlie many other algebraic structures such as fields and vector spaces and are also important tools for studying symmetry in all its forms. For these reasons, group theory is considered to be an important area in modern mathematics, and it has many applications to mathematical physics (for example, in particle theory).

Contents

1 History

2 Basic definitions

3 Notation for groups

4 Some elementary examples and nonexamples

4.1 An abelian group: the integers under addition
4.2 Not a group: the integers under multiplication
4.3 An abelian group: the nonzero rational numbers under multiplication
4.4 A finite nonabelian group: permutations of a set
4.5 Further examples

5 Simple theorems

6 Constructing new groups from given ones

7 Related topics

8 See also

History

See Group theory.

Basic definitions

A group (G, * ) is a nonempty set G together with a binary operation * : G × G → G, satisfying the group axioms. "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).
Identity 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 identity 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 above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms.

It should be noted that there is no requirement in a group that a * b = b * a (commutativity). A group in which this equation holds for all a and b in G, is called abelian (after the mathematician Niels Abel). 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" 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} for 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 : for all 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; · really is a binary operation)
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)
But, it is not true that whenever a is a non-zero integer, there is a non-zero integer b such that ab = ba = 1. For example, a = 2 is a non-zero integer, but no matter what non-zero integer b we choose, |ab| = |2b| ≥ 2 > 1. Note that 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, so the rational numbers form the algebraic structure of a field. 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 abstract 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 xy for the combined action "first do y, then do x"; so that ab 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
aba : RGB → BGR

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

bb = e,
(aba)(aba) = e, and
(ab)(ba) = (ba)(ab) = e;

so each of the above actions has an inverse.

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

(ab)a = a(ba) = aba, and
(ba)b = b(ab) = aba.

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, ab ≠ ba). 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.
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.

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 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 product can also be defined with an infinite number of terms.
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 zero terms. 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.