On groups with Frobenius–Engel elements

A number of properties of periodic and mixed groups with Frobenius–Engel elements are found (Lemmas in Sect. 2 and Theorem 1). The results obtained are used to describe mixed and periodic groups with finite elements saturated with finite Frobenius groups. It is proved that a binary finite group saturated with finite Frobenius groups is a Frobenius group with locally finite complement (Theorem 2). Theorem 3 establishes that in a saturated Frobenius group of a primitive binary finite group $G$ without involutions the characteristic subgroup $\Omega_1(G)$ generated by all elements of prime orders from $G$ is a periodic Frobenius group with kernel $F$ and locally cyclic complement $H$. Moreover, any maximal periodic subgroup $T$ of $G$ is a Frobenius group with kernel $F$ and complement $T\cap N_G(H)$. A number of examples of periodic non-locally finite and mixed groups satisfying Theorem 3 are given.