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#'''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]].
==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.
