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New page: A '''group''' is a 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 '''group''' is a 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
#'''Associativity''': <math>(AB)C = A(BC)</math> where <math>A</math>, <math>B</math> and <math>C</math> are any element of the group
#'''Existence of Identity''': there must exist an identity element <math>I</math> such that <math>IA = AI = A</math>; that is, applying the binary operator to some element <math>A</math> and the identity element <math>I</math> leaves <math>A</math> unchanged
#'''Existence of Inverse''': for each element <math>A</math>, there must exist an inverse <math>A^{-1}</math> such that <math>AA^{-1} = A^{-1}A = I</math>
A group with commutative binary operator is known as [[Abelian group|Abelian]].
[[category|mathematics]]
#'''Closure''': applying the binary operator to any two elements of the group produces a result which itself belongs to the group
#'''Associativity''': <math>(AB)C = A(BC)</math> where <math>A</math>, <math>B</math> and <math>C</math> are any element of the group
#'''Existence of Identity''': there must exist an identity element <math>I</math> such that <math>IA = AI = A</math>; that is, applying the binary operator to some element <math>A</math> and the identity element <math>I</math> leaves <math>A</math> unchanged
#'''Existence of Inverse''': for each element <math>A</math>, there must exist an inverse <math>A^{-1}</math> such that <math>AA^{-1} = A^{-1}A = I</math>
A group with commutative binary operator is known as [[Abelian group|Abelian]].
[[category|mathematics]]