On infinite Frobenius groups

We study the structure of a periodic group G satisfying the following conditions: (F1) The group G is a semidirect product of a subgroup F by a subgroup H; (F2) H acts freely on F with respect to conjugation in G, i.e. for f∈F, h∈H the equality fh=f holds only for the cases f=1 or h=1. In other words H acts on F as the group of regular automorphisms. (F3) The order of every element g∈G of the form g=fh with f∈F and 1≠h∈H is equal to the order of h; in other words, every non-trivial element of H induces with respect to conjugation in G a splitting automorphism of the subgroup F. (F4) The subgroup H is generated by elements of order 3. In particular, we show that the rank of every principal factor of the group G within F is at most four. If G is a finite Frobenius group, then the conditions (F1) and (F2) imply (F3). For infinite groups with (F1) and (F2) the condition (F3) may be false, and we say that a group is Frobenius if all three conditions (F1)-(F3) are satisfied. The main result of the paper gives a description of à periodic Frobenius groups with the property (F4).