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I know an Automorphism is a group G that is an isomorphism where $h:G \rightarrow G$, (G is being mapped to itself) and that Aut(G) is the set of all Automorphisms in G.

I was wondering how I would approach finding all functions in Aut($Z_n$), let's say Aut($Z_4$) for example.

Is it as trivial as showing a bijective map from each element in $Z_4$ to itself?

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if you decide where to map $1$ you have uniquely determined the automorphism, note $1$ needs to be mapped to another generator though.

Also note the composition of two automorphisms $\sigma,\tau$ on $\mathbb Z_n$ sends $1$ to $\tau(1)\times\sigma(1)$, which is why $Aut(\mathbb Z_n)\cong\mathbb Z_n^*$


For the case $\mathbb Z_4$ note $1$ can only be mapped to $1$ or $3$.

this gives us the identity function and the function where $a$ goes to $-a$.

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