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Group (mathematics)
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Group (mathematics)

In mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. 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. Before that 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, including familiar number systems, such as the integers, rational, real, and complex numbers under addition, non-zero rational, real, and complex numbers under multiplication, non-singular matricies under multiplication, invertible functions under composition, and so on. 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 the other algebraic structures such as fieldss 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 has many applications to mathematical physics (for example, in particle theory).

Table of contents
1 History
2 Basic definitions
3 Notation for groups
4 Some elementary examples and nonexamples
5 Simple theorems
6 Constructing new groups from given ones
7 Related topics

History

See Group theory.

Basic definitions

A group (G,*) is defined as a set G together with a binary operation *: G × GG. We write "a * b" for the result of applying the operation * to the two elements a and b of G. To have a group, * must satisfy the following axioms:

You will often also see the axiom The way that the definition above is phrased, this axiom isn't 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.

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 mathematican 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:

However sometimes the group is thought of as analogous to addition and written additively: Usually, only abelian groups are written additively.

When being noncommital, 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:

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:

So we see that not every element of (Z,·) has an inverse and therefore, (Z,·) is not a group. The most we can say is that it is a 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:

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

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

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, abba). 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

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

Constructing new groups from given ones

  1. 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,*).
  2. The product of two groups (G,*) and (H,•) is the set G×H together with the operation (g1,h1)(g2,h2) = (g1*g2,h1h2). The product can also be defined with an infinite number of terms.
  3. 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.
  4. 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.

Related topics

See Glossary of group theory for more definitions in group theory.

See elementary group theory for a list of elementary theorems in group theory.

See List of group theory topics for a list of all group theory topics covered in Wikipedia.

Topics in mathematics related to structure Edit
Abstract algebra | Number theory | Algebraic geometry | Group theory | Monoids | Analysis | Topology | Linear algebra | Graph theory | Universal algebra | Category theory | Order theory