Timeline for If the union of finitely many conjugacy classes hits large enough difference sets, are ther finitely many conjugacy classes?
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| Dec 1, 2020 at 8:09 | comment | added | Ville Salo | To be more explicit: Let $G$ be the infinite dihedral group and apply Theorem 1.1 from [Osin 2010, "Small cancellations ..."], to obtain a $2$-generated group $C$ which contains $G$, has no elements of orders $n \notin \{2, \infty\}$, and has exactly three conjugacy classes. Let $g \in G$ have order $2$ and $s(h) = h^g$ the inner automorphism, which is nontrivial of order $2$ because $g$ is not central (since $G \leq C$ so there are at least two involutions in $C$). Define $H = C \rtimes_s C_2$ where $C_2$ acts by $s$. The property holds, but possibly $H$ has infinitely many conjugacy classes. | |
| Dec 1, 2020 at 7:43 | comment | added | Ville Salo | Maybe if you have not two but finitely many conjugacy classes, you could use an inner automorphism, at least IIRC Osin's construction allows an arbitrary set of orders, and certainly the involutions can't be central. Then again maybe it's the part where $H$ has infinitely many conjugacy classes that is not clear. | |
| Nov 30, 2020 at 18:10 | comment | added | YCor | If $G$ is a group with only 2 conjugacy classes and s is an automorphism of order 2, I see no reason that $H=G\rtimes_s C_2$ to have finitely many conjugacy classes (i.e., whether $G$ has finitely many twisted conjugacy classes, i.e. orbits under $v\cdot u = vusv^{-1}s$). If there's an example, then $H$ would answer negatively the question. But I'm not sure at all it exists (and not sure there's no easier approach). | |
| Nov 30, 2020 at 15:34 | comment | added | Ville Salo | As is standard to ask: why the downvote? | |
| Nov 30, 2020 at 14:15 | history | asked | Ville Salo | CC BY-SA 4.0 |