# Changes

# the set of the positive [[rational number]]s $\mathbb{Q}_+$ under multiplication, $(\mathbb{Q}_+,\cdot)$: $1$ is the identity, while the inverse of an element $\frac{m}{n} \in \mathbb{Q}_+$ is $\frac{n}{m}$.
# for every $n \in \mathbb{N}$ there exists at least one group with n elements,e.g., $(\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+).$
# the set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</i> is the positive principal square root of -1, the basis of the [[imaginary number]]s. This group is [[isomorphism|isomorphic]] to $\mathbb{Z}_{4}$ under mod addition.
# 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.