3.7 (1969)

An automorphism $\sigma$ of a group $G$ is called algebraic if, for any $g \in G$, the minimal $\sigma$-invariant subgroup of $G$ containing $g$ is finitely generated. An automorphism $\sigma$ of $G$ is called an e-automorphism if, for any $\sigma$-invariant subgroups $A$ and $B$, where $A$ is a proper subgroup of $B$, $B \setminus A$ contains an element $x$ such that $[x, \sigma] \in B$. Is every e-automorphism algebraic?