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I'm looking for two things related to the GAP function IsGroupOfAutomorphisms: whether it does what I think it does (based on the brief GAP manual entry), and if so, how it works.

The GAP manual states the following under IsGroupOfAutomorphisms(G): Indicates whether G consists of automorphisms of another group H. The group H can be obtained from G via the attribute AutomorphismDomain (40.7-3).

I assume this means that the function returns true if there exists another group $H$ such that $G\le \text{Aut}(H)$ and returns false otherwise. Can anyone confirm whether this is accurate?

If so, I would be very interested in learning a little about how it works, as such a function would seem like a miracle to me. If you know how the function works or have any resources that deal with the problem I'd love to hear it!

This is my first post here, so please let me know if there are any guidelines I've missed.

Shaun
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Michael Wynne
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    For some groups it is known that they cannot arise as automorphism groups, see for example this post, or others. If GAP knows this, the algorithm is easy for such groups. Return "false" if $G$ is on this list. – Dietrich Burde Jul 08 '24 at 08:22
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    I think you are misunderstanding this function. It returns $\mathbf{true}$ only when it already knows a group $H$ with $G \le {\rm Aut}(H)$, usually because $G$ was defined as a subgroup of the automorphism group of $H$. Otherwise it returns $\mathbf{false}$. So a return value of $\mathbf{false}$ does not mean that $G$ is not equal to (a subgroup of) the automorphism group of some other unknown group, it just means that it does not know of such a group $H$. – Derek Holt Jul 08 '24 at 08:45
  • Thank you both. I figured it was too good to be true :( – Michael Wynne Jul 08 '24 at 08:50
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    Michael, if you're interested in the question "is $G$ isomorphic to some subgroup of an automorphism group of a group?" (the natural group-theoretical way to interpret your proposed meaning) then I think it follows from Cayley's theorem and the fact that symmetric groups embed in their own automorphism groups that this is true for every $G$! Dietrich Burde is talking about the question "is $G$ isomorphic to some automorphism group of a group?", which is different. So the GAP functionality, which is about the actual type of the group elements, is the only sensible meaning for this function. – Izaak van Dongen Jul 08 '24 at 19:32
  • Thanks @IzaakvanDongen , that's a very nice explanation! – Michael Wynne Jul 09 '24 at 01:58

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