Group

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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: (AB)C = A(BC) where A, B and C are any element of the group
  3. Existence of Identity: there must exist an identity element I such that IA = AI = A; that is, applying the binary operator to some element A and the identity element I leaves A unchanged
  4. Existence of Inverse: for each element A, there must exist an inverse A − 1 such that AA − 1 = A − 1A = I

A group with commutative binary operator is known as Abelian.

Example 1: the Klein four group consists of the set of formal symbols {1,i,j,k} with the relations i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j. All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to Z_{2}\times Z_{2} under mod addition.

Example 2: the set of complex numbers {1, -1, i,-i} under multiplication, where i is the square root of -1, the basis of the imaginary numbers. This group is isomorphic to Z4 under mod addition.

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

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