12.62 (1992)

A Frobenius group is a transitive permutation group in which the stabilizer of any two points is trivial. Does there exist a Frobenius group of infinite degree which is primitive as permutation group, and in which the stabilizer of a point is cyclic and has only finitely many orbits?

Conjecture: No. Note that such a group cannot be 2-transitive (Gy. Károlyi, S. J. Kovács, P. P. Pálfy, Aequationes Math., 39 (1990), 161–166; V. D. Mazurov, Siberian Math. J., 31 (1990), 615–617; P. M. Neumann, P. J. Rowley, in: Geometry and cohomology in group theory (London Math. Soc. Lect. Note Ser., 252), Cambridge Univ. Press, 1998, 291–295).