Example 1: Show that the multiplicative group consisting of three cube roots of unity is isomorphic to the group of residue classes under addition of residue classes .
Solution:
Let us consider the composition tables of two structures as given below:
































From this table it is evident that if are replaced by respectively in the composition table for we get the composition table . This leads to the fact that mapping of onto defined by , , is an isomorphism. Also
Example 2: Show that the additive group is an isomorphic to the additive group for any given integer .
Solution:
We define a mapping by , where and show that is an isomorphism of onto .
We see that is oneone since two different elements of have two different image in is the image of an element of .
Again
Thus is composition preserving as well. Hence is an isomorphic mapping of onto .