8.14 (1982)

a) Assume a group $G$ is existentially closed in the class $L\mathfrak{N}_p$ of all locally finite $p$-groups. Is it true that $G$ is characteristically simple? This is true for $G$ countable in $L\mathfrak{N}_p$ (Berthold Maier, Freiburg); in fact, up to isomorphism, there is only one such countable locally finite $p$-group.
b) Assume a group $G$ is existentially closed in one of the classes $L\mathfrak{N}^+$, $L\mathfrak{S}_\pi$, $L\mathfrak{S}^+$, $L\mathfrak{S}$ of, respectively, all locally nilpotent torsion-free groups, all locally soluble $\pi$-groups, all locally soluble torsion-free groups, or all locally soluble groups. Is it true that $G$ is characteristically simple? The existential closedness of $G$ is a local property, thus it seems difficult to obtain global properties of $G$ from it.