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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]].

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

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

Example 2: the set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</i> is the square root of -1, the basis of the [[imaginary number]]s. This group is [[isomorphic]] to <math> Z_{4} </math> under mod addition.

Example 2: the set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</i> is the square root of -1, the basis of the [[imaginary number]]s. This group is [[isomorphism|isomorphic]] to <math> Z_{4} </math> under mod addition.

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

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