# Changes

# the set of [[integers]] $\mathbb{Z}$ under addition: here, zero is the identity, and the inverse of an element $a \in \mathbb{Z}$ is $-a$.
# the set of the positive [[rational numbers]] $\mathbb{Q}_+$under multiplication: obviously, $1$ is the identity, while the inverse of an elements element $\frac{m}{n} \in \mathbb{Q}_+$ is $\frac{n}{m}$.
# 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 [[isomorphism|isomorphic]] to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ 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 $\mathbb{Z}_{4}$ under mod addition.