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If $G$ is a group, we can say $g$ is automorphism-conjugate to $f$ if there is a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. This is an equivalence relation.

Is there a standard name for this equivalence relation? Is it a well-studied notion? Does it have some importance somewhere?

Some alternative ways to state this:

  • like conjugacy classes, we can talk about automorphism orbits in group, and I am just asking for the name of / literature on the related equivalence, or equivalence relation.
  • $f, g \in G$ are automorphism-conjugate if and only if $f$ and $g$ are actually conjugate in the holomorph of $G$.

I feel like I have encountered this before, but I am not even able to find a name in the literature (I found it a little hard to search for because Google confuses it with e.g. conjugacy in automorphism groups, and a range of other things that sound vaguely similar).

I'm thinking about this because I realized that if $G$ is a big homeomorphism group, Rubin's theorem shows that automorphism-conjugacy is a notion between group-theoretic and topological conjugacy (because group automorphisms have to come from topological conjugacies), and these two notions are of great interest to me.

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    $\begingroup$ I don't quite understand. I guess you mean that $g, f$ are fixed group elements here, and that if there is an inner automorphism $\alpha$ mapping $g$ to $f$ then we say that $g$ and $f$ are conjugate -- but what if $\alpha$ is not inner? (I think this is your question). In that case, don't we just say that $g$ and $f$ are in the same automorphic orbit? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Dec 9, 2022 at 11:01
  • $\begingroup$ Yes, that's a possible roundabout way of rewording the definition. $\endgroup$
    – Ville Salo
    Commented Dec 9, 2022 at 11:11
  • $\begingroup$ I agree that this is such a simple notion that it may not need a name. Nevertheless, maybe it has one. Some that I considered are "automorphic", "automorphically conjugate", "automorphism-conjugate", "outer-conjugate" (or something in that direction), "externally/holomorphically conjugate" (the elements are conjugate in the holomorph), some being more sensible than others. I'm fine with also not having a name, but this would help look for information on this. $\endgroup$
    – Ville Salo
    Commented Dec 9, 2022 at 11:15
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    $\begingroup$ In the context of Whitehead's algorithm for free groups, this usually goes by the name "automorphically equivalent" $\endgroup$
    – Robbie Lyman
    Commented Dec 9, 2022 at 15:48
  • $\begingroup$ @LSpice: Yes, I mentioned that above. There are many characterizations of this property. $\endgroup$
    – Ville Salo
    Commented Dec 9, 2022 at 21:27

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