Answer to Question #218159 in Abstract Algebra for PRIYANKA

Question

Any two non-zero subgroups of Z are isomorphic. 

Answer 1

Solution:

Yes, it is correct. Let $H \leq \mathbb{Z}$ be a subgroup. If $H=\{0\}$ then $H=0 \mathbb{Z}$ and we are done. Otherwise H must contain non-zero elements, some of them positive. Let $n=\min \{k>0: k \in H\}$ . Then it can be shown that $H=n \mathbb{Z}$ . We have $n \in H$ and hence obviously $n \mathbb{Z} \leq H$ , because H is closed to addition and additive inverses. For the other direction let $k \in H$ . We can divide k by n with remainder. We get an expression of the form k=q n+r where $q, r \in \mathbb{Z}, 0 \leq r<n$ . But then note that $r=k+(-q) n \in H$ . Since n is by definition the smallest positive number in H and $0 \leq r<n$ we have to conclude that r=0, and hence $k \in n \mathbb{Z}$ . This shows that $H \leq n \mathbb{Z}$ .

If $n \neq 0\ then\ n \mathbb{Z}$ is indeed isomorphic to $\mathbb{Z}$ . We can define an isomorphism $\varphi: \mathbb{Z} \rightarrow n \mathbb{Z}\ by\ k \rightarrow n k$