A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If X is some set of finite groups, then the group 𝐺 saturated with groups from the set X if any finite subgroup of 𝐺 is contained in a subgroup of 𝐺, isomorphic to some group from X. A group 𝐺 = 𝐹⋋𝐻 is a Frobenius group with kernel 𝐹 and a complement 𝐻 if 𝐻 ∩ 𝐻^f = 1 for all 𝑓 ∈ 𝐹^# and each element from 𝐺 ∖ 𝐹 belongs to a one conjugated to 𝐻 subgroup of 𝐺. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Boris Durakov, Postgraduate Student, Siberian Federal University, 79, Svobodny ave., Krasnoyarsk, 660041, Russian Federation, tel.: 8 (391) 206-21-48, email: durakov96@gmail.com

Anatoly Sozutov, Dr. Sci. (Phys.–Math.), Prof., Siberian Federal University, 79, Svobodny ave., Krasnoyarsk, 660041, Russian Federation, tel.: 8 (391) 206-21-48, email: sozutov ai@mail.ru

Durakov B. E., Sozutov A.I. On Periodic Groups Saturated with Finite Frobenius Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 35, pp. 73-86. https://doi.org/10.26516/1997-7670.2021.35.73

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