0

Let $G$ be a group, one might ask if there always exist a group $H$ such that $G\cong\text{Aut}(H)$. The answer for this is no. See this answer for counterexamples, which explains that if $G$ is a finite non-trivial group then an odd cardinality is an obstruction for $G$ to be an automorphism group, or in an alternative sense, that every non-trivial finite automorphism group has even cardinality.

Is there a nice set of necessary conditions $G$ must satisfy for this to be true?

If the general case is too difficult, I'm also accepting answers for finite groups or for inner automorphisms.

Diana Pestana
  • 1,577
  • 1
    Does https://math.stackexchange.com/questions/1747784/groups-which-can-not-occur-as-automorphism-group-of-a-group answer your question? – Gerry Myerson Apr 29 '25 at 07:52
  • It's a similar question, but I'm not looking for obstructions. Quite the opposite, I'm looking for a general set of properties that if a group satisfy, then it's guaranteed to be an automorphism group. I would be even nicer to see if someone can construct such $H$(maybe as a subgroup of $G$, or something else entirely). – Diana Pestana Apr 29 '25 at 20:19
  • The question is a bit soft, as I'm asking for any set of properties. So they can be pretty restrictive, but non-trivial (as saying "it is an automorphism group if it is an automorphism group") – Diana Pestana Apr 29 '25 at 20:26

0 Answers0