What is $\mathbb{Z}_{mn}$ for $m,n \geq1 $? Can someone give an example? What is the relationship between $\mathbb{Z}_{mn}$ and $\mathbb{Z}_m$? What is our 'well-known' group which similar to $\mathbb{Z}_{mn}$?
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$\begingroup$$\mathbb{Z}_{mn}$ should denote the integers modulo $mn$. The relationship between $\mathbb{Z}_m$, $\mathbb{Z}_n$ and $\mathbb{Z}_{mn}$ is:
\begin{equation} \mathbb{Z}_{mn} \cong \mathbb{Z}_m\oplus\mathbb{Z}_n \Leftrightarrow m,n\text{ coprime} \end{equation}
As an example consider $\mathbb{Z}_2$ and $\mathbb{Z}_3$. You can check that $\mathbb{Z}_2\oplus\mathbb{Z}_3\cong \mathbb{Z}_6$ by finding an element in $\mathbb{Z}_2\oplus\mathbb{Z}_3$ of order $6$
$\endgroup$ 4 $\begingroup$I have seen a nice question before about to this question, but cannot find it to make you linked. Anyway, consider the set $\{0,1,2,...,n-1\}$. This set establish a goup under addition in (mod n). More over this group is cyclic and it has $n$ elements. We always denote it by $\mathbb Z_n$. So $$\mathbb Z_n=\{0,1.2...,n-1\}$$. From above you can easily find out what is $\mathbb Z_{mn}$. For the second question, I am adding an exampe in which you can understand the relation between $\mathbb Z_{mn}$ and for example $\mathbb Z_m$. Indeed: $$\mathbb Z_m\cong H\leq\mathbb Z_{mn}$$ when $(m,n)=1$. In fact, there is a theorem says this property as @Daniel noted. I hope you can find the difference between $n\mathbb Z$ and $\mathbb Z_n$ as well.
$\endgroup$ 7 $\begingroup$The most pertinent relation between $\def\Z{\Bbb Z}\Z_{mn}=\Z/mn\Z$ and $\Z_m$ is that there is a (unique) ring morphism $\Z_{mn}\to\Z_m$, in other words, if you know the remainder of some integer modulo $mn$, then you can also determine its remainder modulo $m$ (and arithmetic operations modulo $mn$ correspond to the same operations after having reduced modulo $m$). It is important to note that such a morphism $\Z_k\to\Z_m$ exists only if $k$ is a multiple of $m$; in other cases representatives of the same remainder modulo $k$ will in general have different remainders modulo $m$, making it impossible to define such a morphism.
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