The locally quaternion group is a countable $B$-group

Abstract

A group $G$ is a $B$-group if every primitive permutation group containing a regular subgroup isomorphic to $G$ is doubly transitive. We prove that the locally quaternion group $\mathrm{Dic}(C_{2^\infty},c)$, the extension of the Prüfer $2$-group by an element squaring to its involution and inverting it, is a $B$-group. This gives an example of a countably infinite $B$-group. This answers Problem 13.30 of the Kourovka Notebook. The proof analyses the automorphism group of an orbital Cayley graph through the central involution. In the main case two half-flip automorphisms generate, with their conjugates, an elementary abelian normal subgroup whose orbits form a nontrivial block system; in the remaining case, where the pointwise stabilizers of the two cosets of the Prüfer group have order two, the conjugates of the central translation generate an elementary abelian normal subgroup, which would be regular if the automorphism group were primitive; an affine lemma for the Prüfer group excludes this. The proof does not depend on computer verification: its two finite case analyses are written out in full, and programs that re-check them are provided as ancillary files.