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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2024 Volume 30, Number 1, Pages 213–222 (Mi timm2073)

On groups with Frobenius–Engel elements

A. I. Sozutov

Siberian Federal University, Krasnoyarsk

Abstract: 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.

UDC: 512.54

MSC: 20E25

Received: 18.10.2023
Revised: 01.02.2024
Accepted: 05.02.2024

DOI: 10.21538/0134-4889-2024-30-1-213-222



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© Steklov Math. Inst. of RAS, 2025