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.
Keywords: Frobenius groups, finite elements, Engel elements, Frobenius elements, Frobenius-Engel elements, saturation
Received October 18, 2023
Revised February 1, 2024
Accepted February 5, 2024
Funding Agency: This work was supported by the Russian Science Foundation (project no. 19-71-10017).
Anatoly Ilich Sozutov, Dr. Phys.-Math. Sci., Prof., Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: sozutov_ai@mail.ru
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Cite this article as: A.I. Sozutov. On groups with Frobenius–Engel elements. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 213–222.
