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Ville Salo
Ville Salo
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Suppose I have a groupIf $G$ is a group, and I havewe can say $g$ is automorphism-conjugate to $f$ if there is a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. In case $\alpha$ is inner thisThis is standard conjugacy. In general it's some kind of natural notion of similarityan equivalence relation.

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

Is there a standard name? 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.

Suppose I have a group $G$, and I have a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. In case $\alpha$ is inner this is standard conjugacy. In general it's some kind of natural notion of similarity.

Is there a standard name? Is it a well-studied notion? Does it have some importance somewhere? 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.

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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Martin Sleziak
Martin Sleziak
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Ville Salo
Ville Salo
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Automorphism-conjugacy

Suppose I have a group $G$, and I have a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. In case $\alpha$ is inner this is standard conjugacy. In general it's some kind of natural notion of similarity.

Is there a standard name? Is it a well-studied notion? Does it have some importance somewhere? 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.