Question on automorphism groups

Hello Prof Kim

I got a question about groups and ring ( algebra 4). Can you explain to me what it means to “describe $Aut(C_n)$ explicitly” for n = 1, 2,3, ….?

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In many parts of mathematics, one uses the notation $Aut(S)$ to denote the automorphisms of $S$, where $S$ is a set usually with some extra structure. Thus, $Aut(S)$ consists of maps

$f:S \rightarrow S$

with the property that

(1) $f$ has an inverse;

(2) $f$ is compatible with whatever structure $S$ has, if any.

So when you’re being asked to describe, say, $Aut(C_3)$ explicitly, you are being asked to describe all invertible maps

$f: \{0,1,2 \} \rightarrow \{0,1,2\}$

that are compatible with the group structure. One point you shouldn’t get confused by is that $Aut(S)$ isl *always* a group under composition, whatever structure $S$ has. If $S$ is itself a group, $Aut(S)$ will in general be some other group.

Note that if we had left out the group structure and just considered the set $\{0,1,2\}$, then the automorphisms would be isomorphic to (‘essentially the same as’) $S_3$, the symmetric group on three letters. But with the group structure, you need to be more careful. Not all permutations will preserve the group structure. Think about it a bit and ask again if it’s still confusing.