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Line 8: − 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 [[isomorphism|isomorphic]] to <math> Z_{4} </math> under mod addition.+ + +
simpler examples added
A '''group''' is a mathematical structure consisting of set of elements combined with a binary operator which satisfies four conditions:
A '''group''' is a mathematical structure consisting of 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
#'''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]].
==Examples==
# the set of integers <math>\mathbb{Z}</math> under addition: here, zero is the identity, and the inverse of an element <math>a \in \mathbb{Z}</math> is <math>-a</math>.
# the set of the positive rational numbers <math>\mathbb{Q}_+</math>: obviously, <math>1</math> is the identity, while the inverse of an elements <math>\frac{m}{n} \in \mathbb{Q}_+</math> is <math>\frac{n}{m}</math>.
# 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>\mathbb{Z}_{2} \times \mathbb{Z}_{2}</math> under mod addition.
# 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> \mathbb{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]].