19.29 (2018)

Let $\Phi \in \text{Out}\,G = \text{Aut}\,G/\text{Inn}\,G$. Two automorphisms $\varphi, \psi \in \Phi$ are said to be isogradient if $\varphi = \hat{g}^{-1} \psi \hat{g}$ for some inner automorphism $\hat{g}$. The number of isogradiency classes in $\Phi$ is denoted by $S(\Phi)$.

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