Question: are z and 2z isomorphic?
Answer: Example 18 Let Z be the integers under addition and let 2Z be the set of even integers under addition. The function / : Z ( 2Z is an isomorphism. Thus Z 'φ 2Z. (Thus note that it is possible for a group to be isomorphic to a proper subgroup of itself Pbut this can only happen if the group is of infinite order).