Abstract Algebra: Theory and Applications

Many groups may appear to be different at first glance, but can be shown to be the same by a simple renaming of the group elements. For example, ${\mathbb{Z}}_4$ and the subgroup of the circle group $\mathbb{T}$ generated by $i$ can be shown to be the same by demonstrating a one-to-one correspondence between the elements of the two groups and between the group operations. In such a case we say that the groups are isomorphic.