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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Reply:

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.

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