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Let $n$Z be the set of integer multiples of $n \in$ Z.

Can someone explain how Z$/n$Z is isomorphic to Z$_n$?

Specifically, what is the function that establishes the isomorphism and how can we be sure it is bijective?

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1 Answer

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Presumably we can denote the elements of $\Bbb Z_n$ as $[a]_n$ where $a \in \Bbb Z$ and $[a]_n$ denotes the equivalence class of $a$ modulo $n$.

On the other hand, the elements of $\Bbb Z/n\Bbb Z$ can be written as cosets of the form $a + n\Bbb Z$.

We can define an isomorphism between these groups by the map $$ [a]_n \mapsto a + n\Bbb Z $$

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Note, however, that $[a]_n$ and $a + n\Bbb Z$ are equal as sets and the group operation is precisely the same, so the isomorphism of these two groups is fairly trivial.

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