Timeline for Infinite groups with oligomorphic conjugation action
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2021 at 21:18 | comment | added | Dima Pasechnik | non-existence of such groups was proved in mathscinet.ams.org/mathscinet-getitem?mr=2194251 using a classification of locally finite groups (in turn, relying on CFSG). Cherlin, Gregory (1-RTG); Djordjevic, Marko (S-UPPS); Hrushovski, Ehud (IL-HEBR-IM) A note on orthogonality and stable embeddedness. (English summary) J. Symbolic Logic 70 (2005), no. 4, 1359–1364 | |
| Apr 19, 2017 at 17:50 | answer | added | Guntram | timeline score: 8 | |
| Apr 12, 2017 at 22:12 | comment | added | Ben Wieland | $(\mathbb Z/p)^\infty$ has action by $Aut(G)$ oligomorphic. Remaining questions: 1. Is there an infinite fg group such that $Aut(G)$ acts on $G^2$ with finitely many orbits? 2. Is there an infinite (not necessarily fg) group such that conjugation acts on $G^2$ with finitely many orbits? | |
| Apr 10, 2017 at 6:35 | vote | accept | Andreas Thom | ||
| Apr 9, 2017 at 22:22 | comment | added | YCor | Another trivial remark: the $G$-action on $G^n$ has finitely many orbits if and only if the $G\times G$-action (by left-and-right translation) on $G^{n+1}$ has finitely many orbits. | |
| Apr 9, 2017 at 21:24 | answer | added | YCor | timeline score: 12 | |
| Apr 8, 2017 at 1:10 | comment | added | user6976 | It is my first reaction. | |
| Apr 7, 2017 at 21:48 | comment | added | YCor | @MarkSapir is this an expectation of yours? a claim? | |
| Apr 7, 2017 at 6:17 | history | edited | Andreas Thom | CC BY-SA 3.0 |
added 171 characters in body
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| Apr 7, 2017 at 0:45 | comment | added | user6976 | Conjugation 3-oligomorphic groups should be finite. | |
| Apr 6, 2017 at 23:12 | comment | added | Gerry Myerson | I had a theorem about oleomorphic conjugation, but the margarine was too small to contain it. | |
| Apr 6, 2017 at 21:06 | comment | added | Julien Melleray | Any conjugation-oligomorphic group must be $\aleph_0$-categorical, and these groups are well-studied, I think (and are easily seen to be uniformly locally finite, as Yves points out). This comment is just here to point out that you might want to look into literature on $\aleph_0$-categorical groups... | |
| Apr 6, 2017 at 20:37 | comment | added | YCor | Also, if $G$ is conjugation-oligomorphic and infinite, it has finitely many normal subgroups. In particular, the intersection $N$ of finite index subgroups has finite index, hence $N$ is also conjugation-oligomorphic and infinite, and in addition has no proper subgroup of finite index. Since max-n passes to finite index, $N$ has a simple quotient. In conclusion, if there exists a conjugation-oligomorphic infinite group, then there is a simple one. | |
| Apr 6, 2017 at 20:13 | comment | added | YCor | Besides, clearly any conjugation-oligomorphic group is locally finite. Indeed, if $\gamma_1,\dots,\gamma_n$ are any elements and generate a subgroup $H$, then the $(\gamma_1,\dots,\gamma_n,x)$ for $x\in H$ are in distinct orbits of the $G$-action (and even $\mathrm{Aut}(G)$-action) on $G^{n+1}$. Hence a finitely generated conjugation-oligomorphic group is finite. | |
| Apr 6, 2017 at 18:40 | comment | added | YCor | Already assuming that $G$ has finitely many orbits on $G^2$ by conjugation implies that $G$ is of bounded torsion. I guess it's open to find an infinite $G$ with finitely many orbits on $G^2$. | |
| Apr 6, 2017 at 18:26 | history | edited | Andreas Thom | CC BY-SA 3.0 |
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| Apr 6, 2017 at 15:01 | history | asked | Andreas Thom | CC BY-SA 3.0 |