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.

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:

Related Topics:
Nonempty - Set - Binary operation - Axiom

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• 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

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• 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.

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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.

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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.

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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.

Related Topics:
Commutative - Abelian - Niels Abel

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The order of a group G, denoted by |G| or o(G), is the number of elements of the set G.

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A group is called finite if it has finitely many elements, that is if the set G is a finite set.

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Note that we often refer to the group (G, * ) as simply "G", leaving the operation * unmentioned.

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But to be perfectly precise, different operations on the same set define different groups.

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