# Infinite group with finitely many conjugacy classes of cardinality $n$.

Does there exist an infinite group $$G$$ such that:

1. There are no conjugacy classes containing infinitely many elements.
2. For every $$n \in \mathbb{N}$$, there are only finitely many conjugacy classes containing exactly $$n$$ elements.

Some basic observations:

• $$G$$ cannot be Abelian, otherwise it would have infinitely many conjugacy classes containing $$1$$ element.
• $$G$$ must have infinitely many conjugacy classes.

A basic idea I had was to construct a group $$G := \bigoplus_{n \in \mathbb{N}} G_n,$$ where $$G_n$$ is a finite group with $$2$$ conjugacy classes: one containing the neutral element, of size $$1$$, and the other containing all other elements, of size $$p_n$$. If all $$p_n$$ are prime and $$p_1 < p_2 < \cdots < p_n < \cdots$$, I believe the conditions would be satisfied. However, I have no idea if there are infinitely many primes $$p_n$$ for which such groups $$G_n$$ exist...

• For starters, a finite group with only $2$ conjugacy classes must be cyclic of order $2$. – Nicky Hekster Sep 6 '19 at 16:40
• The group would be "very abelian", in that every centralizer will have finite index... – Arturo Magidin Sep 6 '19 at 20:36
• Actually, finite groups with $\le m$ conjugacy classes have order $\le c_m$ for some $c_m$, see groupprops.subwiki.org/wiki/…. This is an elegant application of the fact that every positive rational has only finitely many representations as a sum of $k$ inverses of (possibly equal) positive integers for each given $k$ ("Egyptian fraction representations"). – YCor Sep 7 '19 at 8:48

Just take a direct sum $$G=\bigoplus G_n$$ where in $$G_n$$ the smallest size $$c_n$$ of a conjugacy class $$\neq\{1\}$$ satisfies $$c_n\to\infty$$. For instance $$G_n$$ the symmetric group works (in $$S_n$$ for $$n\ge 3$$ every conjugacy class has cardinal $$\ge n$$, and actually much more). Alternatively take prime powers $$q_n\to\infty$$ and the affine group $$G_n=\mathbf{F}_{q_n}\rtimes \mathbf{F}_{q_n}^*$$, in which every nontrivial conjugacy class has cardinal $$\ge q_n-1$$. This gives a solvable (metabelian) example.
Indeed if $$g$$ has a conjugacy class of size $$c$$ and $$c_n>c$$ for $$n\ge n_0$$, then $$g\in\bigoplus_{n, which leaves finitely many possibilities for $$g$$.