Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations

Let $\mathfrak{X}$ be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive $\mathfrak{X}$-groups are countably recognizable, while totally imprimitive $\mathfrak{X}$-groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive $\mathfrak{X}$-subgroups. It turns out that totally imprimitive $p$-groups in the class $\mathfrak{X}$ are countably recognizable.