Group (mathematics)

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.

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.

Related Topics:
Integer - Addition

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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.

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Then (Z,+) is a group (written additively).

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Proof:

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• 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.

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The integers with both addition and multiplication together form the more complicated algebraic structure of a ring.

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In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

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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:

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• 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 a 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.

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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

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a and b are integers and b is nonzero, and the operation multiplication, denoted by "·".

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Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group.

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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).

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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.

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Just as the integers form a ring, so the rational numbers form the algebraic structure of a field.

Related Topics:
Ring - Field

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

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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".

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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".

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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:

Related Topics:
Permutation - Set

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• 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.

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Similarly,

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

so each of the above actions has an inverse.

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

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

This group is called the symmetric group on 3 letters, or S3.

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It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ ba).

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Since S3 is built up from the basic actions a and b, we say that the set {a,b} generates it.

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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.

Related Topics:
Permutation group - Cayley's theorem - Group action

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Further examples

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

Related Topics:
Examples of groups - List of small groups

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