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this is my first time on the Stack Exchange so I apologize if I went about asking this improperly! I was working on a problem set that asked to compute Aut(S3). In class we've been thinking about automorphisms as a function that maps elements of one group to another. However it seems like an automorphism can be better thought of as a subgroup of S3. Could someone provide clarity about what it means to compute/find Aut(S3)?

Josh Charleston
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  • Automorphism is a function from the group to itself that respects the operation. Do you have a representation for $S_3$? Probably that helps. – Phicar Jul 22 '20 at 20:25
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    Can you explain how you think an automorphism could be thought of as a subgroup? I think that will help you get an answer that clears up your misunderstanding. – Matthew Towers Jul 22 '20 at 20:56
  • The automorphisms of a group $G$ are particular (group's operation preserving) bijections on $G$, so the only you may hope -in terms of subgroups- is that $\operatorname{Aut}(G)\le \operatorname{Sym}(G)$ (which is indeed the case). –  Jul 22 '20 at 21:09
  • @MatthewTowers I think an automorphism is a bijective isomorphism from a group to itself. So when asking for Aut(S3) it's asking for the group containing all bijective isomorphisms of S3. Is my understanding now correct? – Josh Charleston Jul 22 '20 at 22:24
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    Specifically for $\operatorname{Aut}(S_3)$, see this: https://math.stackexchange.com/q/1538681/810157 –  Jul 23 '20 at 06:38
  • The phrase "thinking about automorphisms as a function" doesn't read well and perhaps illustrates an issue you have. The phrase mixes "automorphisms" (plural) and "function" (singular). The correct form is: automorphisms (plural) are functions (plural), and they form a group. An automorphism (singular) is a function (singular), and it does not in general form a group. – user1729 Jul 23 '20 at 07:27

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The automorphism group $\operatorname{Aut} S_3$ is the group of automorphisms on $S_3$. So it’s a group, but its elements are automorphisms of $S_3$,

$$\operatorname{Aut} S_3 = \{σ \colon S_3 → S_3;~σ~\text{is a group automorphism} \}.$$

Keep in mind that automorphisms are meant to be maps from a group to itself – and not to another group. It stems from the greek word αὐτóς, meaning “self” or “same”, so its something that shapes (grk. “μορφóω”) the group into itself, the same group. Automobiles are things that movable (lat. “mobilis”) by themselves. Most terms in mathematics actually have meaning.

k.stm
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  • Also it’s not too hard to prove that for any group $G$, Aut($G$) is a group under function composition. The identity is the automorphism that maps all elements in $G$ to themselves; automorphisms are isomorphisms, hence they’re bijective and invertible, hence every element has an inverse; function composition is associative. Done. – Ty Jensen Jul 22 '20 at 22:01
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I think this

In class we've been thinking about automorphisms as a function that maps elements of one group to another.

is not quite right. Homomorphisms are functions that map the elements of one group to another (and preserve the group multiplication).

When the domain and codomain of a homomorphism are the same group and the homomorphism is bijective (so an isomorphism) then you call that homomorphism an automorphism.

If you have a group $G$ (say, $S_3$) then you can consider all the automorphisms of that group. The set of those automorphisms is itself a group (under functional composition) called the automorphism group of $G$.

You have been asked to find that group - start by listing its elements.

.

Ethan Bolker
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