Timeline for Infinite groups with 2 automorphism orbits
Current License: CC BY-SA 4.0
14 events
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| Feb 26 at 4:38 | comment | added | Glasby | The locally compact abelian groups $G$ where ${\rm Aut}(G)$ has at most 2 topological orbits on $G$ are described in Theorem 2.13 of Markus Stroppel, Locally compact groups with few orbits under automorphisms. Topology Proc. 26 (2001/02), no. 2, 819–842. | |
| Feb 18 at 1:17 | comment | added | Denis T | On the other hand, there are quite beautiful constructions by J. Berrick of universal embeddings of groups into perfect groups. It is called "binate tower", and is a special case of HNN tower mentioned by Moishe Cohan earlier. | |
| Feb 18 at 1:13 | comment | added | Denis T | I think that the task of classification of finitely generated groups with 2 conjugacy classes is already completely intractable: Denis Osin's construction of a f. g. group with 2 conjugacy classes depends on a lot of arbitrary choices. | |
| Feb 18 at 0:58 | history | edited | Glasby | CC BY-SA 4.0 |
clarified Edit (introduced the field name $F$).
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| Feb 17 at 18:44 | comment | added | YCor | I should add that this class includes no infinite locally finite group. Indeed, this should then be a infinite $p$-group (of exponent $p$), but on the other hand it is known that no infinite locally finite simple group is locally solvable (see e.g. Corollary 3.5 here — Locally finite, simple groups by Ulrich Meierfrankenfeld). | |
| Feb 17 at 11:28 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Feb 17 at 10:17 | comment | added | Moishe Kohan | You can start with any torsion-free group and embed it in an infinite group with exactly two conjugacy classes using a countably-infinite chain of HNN extensions. Given how flexible this construction is, I think there is no hope for any reasonable classification. | |
| Feb 17 at 10:16 | history | edited | Glasby | CC BY-SA 4.0 |
fixed typo
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| Feb 17 at 10:06 | history | edited | Glasby | CC BY-SA 4.0 |
added 252 characters in body
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| Feb 17 at 9:36 | comment | added | Glasby | @JeremyRickard Thank you. I corrected the typos. | |
| Feb 17 at 9:33 | history | edited | Glasby | CC BY-SA 4.0 |
added 21 characters in body
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| Feb 17 at 9:27 | comment | added | YCor | Easy fact: An abelian group satisfies this property (2 orbits under Aut) iff it is a simple module over its automorphism group, iff it is a nonzero vector space over either $\mathbf{Q}$ or $\mathbf{F}_p$ for some prime $p$ (no need to go through division algebras to see this). Nonabelian examples are perfect groups and have no proper subgroup of finite index. | |
| Feb 17 at 9:16 | comment | added | Jeremy Rickard | The sentence after Question 1 has a few typos, and I'm not quite sure that I understand what you meant to write. In particular, when you say "The additive group", do you mean "The automorphism group of the additive group"? So are you just pointing out that the additive group of a division algebra is a 2-orbit group? The reason that I think this might not be what you meant is that then it's not clear why you mentioned division algebras, as the additive group of any nonzero vector space is a 2-orbit group. | |
| Feb 17 at 7:22 | history | asked | Glasby | CC BY-SA 4.0 |