I think the following proof should be sufficient to show that $\mathbb Z$ and $6 \mathbb Z$ are isomorphic, but I am not sure.
The bijective map $\phi(g): \mathbb Z \rightarrow 6 \mathbb Z$ is $\phi(g) = g^{6}$, where $g \in \mathbb Z$.
$\phi(g_1 \cdot g_2) = (g_1 \cdot g_2)^{6} = (g_1)^{6} \cdot (g_2)^{6} = \phi(g_1) \cdot \phi(g_2) = (g_1)^6(g_2)^6$ where $g_1,g_2 \in \mathbb Z$