Difference between revisions of "Group (mathematics)"

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A '''group''' is a set of elements combined with a binary operator which satisfies four conditions:
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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
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A group with commutative binary operator is known as [[Abelian group|Abelian]].
 
A group with commutative binary operator is known as [[Abelian group|Abelian]].
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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]].
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Groups are the appropriate mathematical structures for any application involving symmetry.
  
 
[[Category:Algebra]]
 
[[Category:Algebra]]

Revision as of 13:13, 29 April 2007

A group is a mathematical structure consisting of set of elements combined with a binary operator which satisfies four conditions:

  1. Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
  2. Associativity: where , and are any element of the group
  3. Existence of Identity: there must exist an identity element such that ; that is, applying the binary operator to some element and the identity element leaves unchanged
  4. Existence of Inverse: for each element , there must exist an inverse such that

A group with commutative binary operator is known as Abelian.

Example: the Klein Four Group consists of the set of numbers {1, -1, i,-i} under the binary operation of multiplication, where i is the square root of -1, the basis of the imaginary numbers.

Groups are the appropriate mathematical structures for any application involving symmetry.