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This article is cited in 6 scientific papers (total in 6 papers)
Algebraic and logical methods in computer science and artificial intelligence
On periodic groups saturated with finite Frobenius groups
B. E. Durakov, A. I. Sozutov Siberian Federal University, Krasnoyarsk, Russian Federation
Abstract:
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 $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ 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.
Keywords:
Frobenius group, weakly conjugate biprimitive finite group, locally finite radical, saturation condition.
Received: 30.12.2020
Citation:
B. E. Durakov, A. I. Sozutov, “On periodic groups saturated with finite Frobenius groups”, Bulletin of Irkutsk State University. Series Mathematics, 35 (2021), 73–86
Linking options:
https://www.mathnet.ru/eng/iigum445 https://www.mathnet.ru/eng/iigum/v35/p73
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