Group (mathematics)

The mathematical concept of a group is one of the fundamental notions of modern algebra. Groups underlie the other algebraic structures such as fields and vector spaces and are also important tools for studying symmetry in all its forms. The branch of mathematics which study groups is called the group theory.

Table of contents 1 Basic definitions 2 Notation for groups 3 Some elementary examples and nonexamples 3.1 An abelian group: the integers under addition 3.2 Not a group: the integers under multiplication 3.3 An abelian group: the nonzero rational numbers under multiplication 3.4 A finite nonabelian group: permutations of a set 3.5 Further examples 4 Constructing new groups from given ones 5 Related topics

Basic definitions

A group (G,*) is defined as a set G together with a binary operation *: G × G → G. 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:

• 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 = a * e.
• Inverse element: For all a in G, there is an element b in G such that a * b = e = b * a, 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 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:

• 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 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.

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:

• 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 = a + 0. (Identity element)
• If a is an integer, then there is an integer b := −a, such that a + b = 0 = b + a. (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 = a · 1. (Identity element)
• But, if a is an integer, there is not necessarily an integer b such that a · b = 1 = b · a. There may be a rational number b like that, but not an integer. (Inverse element fails)

So we see that not every element of (Z,·) has an inverse and therefore, (Z,·) is not a group.

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. (Remember to also verify that · is a binary operation on Q \ {0} by checking closure.)

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.

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 direct sum of two groups (G,*) and (H",•) is the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂).
3. 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.