19.28 (2018)

Let $G$ be a group, and $\phi$ an automorphism of $G$. Elements $x, y \in G$ are said to be $\phi$-conjugate if $x = z^{-1} y \phi(z)$ for some $z \in G$. The $\phi$-conjugacy is an equivalence relation; the number of $\phi$-conjugacy classes is denoted by $R(\phi)$.

Conjecture: If a finitely generated residually finite group $G$ has an automorphism $\phi$ such that $R(\phi)$ is finite, then $G$ has a soluble subgroup of finite index.

The conjecture is proved if $\phi$ has prime order (E. Jabara, J. Algebra, 320, no. 10 (2008), 3671–3679). If $G$ is infinitely generated, then $G$ does not have to be almost solvable (K. Dekimpe, D. Gonçalves, Bull. London Math. Soc., 46, no. 4 (2014), 737–746), but if it is of finite upper rank, then it has to be almost solvable (E. Troitsky, J. Group Theory, 28, no. 1 (2025), 151–164).