Let G be an infinite group wich is finitely generated.


Is that true that the size of all finite conjugacy classes is bounded?


What I know. If G is a finitely generated FC-group then it's true (follows from [this](https://groupprops.subwiki.org/wiki/Finitely_generated_and_FC_implies_FZ)). But if G isn't FC- or FZ-group, so it's center has an infinite index, then I don't see any problems to have conjugacy classes which sizes tends to infinity. But also I don't see any examples of it :-)


On the other hand Osin and Hull proved (see [here](https://arxiv.org/pdf/1107.1826.pdf)) that there exists groups with finite number of conjugacy classes and that number of conjugacy classe may obey to almost any function. So it looks like an argument for existence a group which finite conjugacy classes are not bounded.



Size of conjugacy classes in infinite groups
The answer is no: there exists a 2-generated group, having finite conjugacy classes of unbounded size.


Indeed B.H. Neumann (1937) produced a 2-generated group $G$ with normal subgroups $(H\_n)\_{n\ge 5}$ such that $H\_n\simeq \mathrm{Alt}\_n$. Since $H\_n$ has a conjugacy class of size growing to infinity (say, the set of 3-cycles), so does $G$ (its conjugacy class in $G$ is contained in $H\_n$, hence is finite too — actually it's unchanged in $G$, if one looks at the construction).


