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In mathematics and abstract algebra, a group is the algebraic structure , where G is a nonempty set and denotes a binary operation called the group operation. The notation is normally shortened to the infix notation , or even to xy.
A group must obey the following rules (or axioms). Let a,b,c be arbitrary elements of G. Then:
 A1, Closure. . This axiom is often omitted because a binary operation is closed by definition.
 A2, Associativity. .
 A3, Identity. There exists an identity (or neutral) element such that . The identity of G is unique by Theorem 1.4 below.
 A4, Inverse. For each , there exists an inverse element such that . The inverse of a is unique by Theorem 1.5 below.
An abelian group also obeys the additional rule:
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