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A '''group''' is a set of elements combined with a binary operator which satisfies four conditions:
A '''group''' is a mathematical structure consisting of set of elements combined with a binary operator which satisfies four conditions:
#'''Closure''': applying the binary operator to any two elements of the group produces a result which itself belongs to the group
#'''Closure''': applying the binary operator to any two elements of the group produces a result which itself belongs to the group
A group with commutative binary operator is known as [[Abelian group|Abelian]].
A group with commutative binary operator is known as [[Abelian group|Abelian]].
Example: the Klein Four Group consists of the set of numbers {1, -1, <i>i</i>,<i>-i</i>} under the binary operation of multiplication, where <i>i</i> is the square root of -1, the basis of the [[imaginary numbers]].
Groups are the appropriate mathematical structures for any application involving symmetry.
[[Category:Algebra]]
[[Category:Algebra]]