Hello Prof Kim

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

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Reply:

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

with the property that

(1) has an inverse;

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

So when you’re being asked to describe, say, explicitly, you are being asked to describe all invertible maps

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

Note that if we had left out the group structure and just considered the set , then the automorphisms would be isomorphic to (‘essentially the same as’) , 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.