Isomorphisms and integers

Question

One more question for me. Is it incorrect to say that (m$Z$,+) is isomorphic to (n$Z$,+) because both are infinite cyclic groups that are isomorphic Z under addition?

Answer 1

The property of two groups being isomorphic is an equivalence relation, and in particular it is transitive. That is to say, if $G_1 \cong G_2$ and $G_2 \cong G_3$, then $G_1 \cong G_3$.

To be specific, imagine you have an isomorphism $\phi:G_1 \rightarrow G_2$ and an isomorphism $\psi:G_2 \rightarrow G_3$. Then you can check that $\psi \circ \phi:G_1 \rightarrow G_3$ is an isomorphism.

Answer 2

The statement

For all $m,n\in\mathbb{Z}$, $(m\mathbb{Z},+)$ is isomorphic to $(n\mathbb{Z},+)$

is false. Indeed, in the case $m=1$, $n=0$ the two groups are not isomorphic.

The statement

For all $m,n\in\mathbb{Z}$, if $m\ne0$ and $n\ne0$, then $(m\mathbb{Z},+)$ is isomorphic to $(n\mathbb{Z},+)$

is true and your argument is correct: both are isomorphic to $(\mathbb{Z},+)$; if $m\ne0$, the map $x\mapsto mx$ is an isomorphism from $(\mathbb{Z},+)$ to $(m\mathbb{Z},+)$. The inverse map of an isomorphism is again an isomorphism and the composition of isomorphisms is an isomorphism.