Groups saturated with finite Frobenius groups
type: Item
instance of: scholarly article
title: Groups saturated with finite Frobenius groups
zbMATH Open document ID: 1478.20030
DOI: 10.1134/S0001434621010314
author: Yanyan Li
published in: Mathematical Notes
publication date: 2021-07-08T00:00:00Z
review text: A group \(G\) is saturated with groups in a set \(\mathfrak{X}\) of groups if every finite subgroup of \(G\) is contained in a subgroup of \(G\) isomorphic with some group in \(\mathfrak{X}\). An infinite group \(G\) is called a Frobenius group with complement \(H\) and kernel \(F\) if \begin{itemize} \item[(1)] \(F,H\) are proper subgroups of \(G\) such that \(G=FH\), \(F\triangleleft G\) and \(F\cap H=1\); \item[(2)] \((G,H)\) is a Frobenius pair, so \(H\cap H^g=1 \forall g\in G\setminus H\); \item[(3)] \(G\setminus (F\setminus 1)=\cup_{g\in G} H^g\). \end{itemize} An element \(a\) in a group \(G\) is finite if all subgroups of the form \(\langle a, a^g\rangle\) are finite for all \(g\in G\). Theorem 1: Let \(G\) be a periodic group with nontrivial locally finite radical \(R\). Suppose that \(G\) is saturated with finite Frobenius groups and contains a finite non-Engel element \(a\) of prime order. If \(a\in R\), then \(G=F\rtimes H\) is a Frobenius group with kernel \(F<R\) and complement \(H\) where \(a\in H\) and either \(H=N_G(\langle a\rangle)\) or \(\langle a^H\rangle\cong \mathrm{SL}_2(3) \text{or } \mathrm{SL}_2(5)\). If \(a\notin R\), then \(G=F\rtimes H\) where \((G,H)\) is a Frobenius pair, \(a\in H\) and \(F\rtimes \langle a\rangle\) is a Frobenius group with kernel \(F>R\) and complement \(\langle a\rangle\). A group \(G\) is called a Shunkov group if for any finite subgroup \(H\) of \(G\) any two conjugate elements of prime order in \(G\) generate a finite group in the quotient \(N_G(H)/H\). Theorem 2: Let \(G\) be a Shunkov group and suppose that the locally finite radical is nontrivial and is saturated with Frobenius groups. Then the set of all elements of \(G\) of finite order in \(G\) is a subgroup \(T\) of \(G\). Furthermore, \(T\) is a Frobenius group with locally finite complement.
reviewed by: Martyn Dixon
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 7369701
zbMATH Keywords: Frobenius group
zbMATH Keywords: Shunkov groups
zbMATH Keywords: saturation
zbMATH Keywords: Engel element
zbMATH Keywords: finite element
MaRDI profile type: Publication
cites work: On certain torsion groups saturated with finite simple groups
cites work: ON A GENERALIZATION OF FROBENIUS' THEOREM TO INFINITE GROUPS
cites work: Q3270631
cites work: Infinite groups saturated by Frobenius subgroups
cites work: Infinite groups saturated with Frobenius subgroups. II
cites work: Q4242910
cites work: On the periodic part of some Shunkov groups
cites work: On some groups with finite involution saturated with finite simple groups.
cites work: Periodic groups saturated with L 3(2m)
cites work: Groups containing a strongly embedded subgroup.
cites work: Groups saturated by finite simple groups.
cites work: Sylow 2-subgroups of the periodic groups saturated with finite simple groups
cites work: Periodic groups saturated with finite simple groups of types \(U_3\) and \(L_3\)
cites work: Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic
cites work: 2-rank two periodic groups saturated with finite simple groups
cites work: Q3084921
cites work: The set of elements of finite order in a biprimitively finite group
cites work: Products of locally dihedral subgroups.
cites work: On Frobenius groups
cites work: On the Shunkov groups acting freely on Abelian groups.
cites work: Periodic groups acting freely on abelian groups.
cites work: Periodic products of groups
cites work: Q4002122
cites work: Groups with Frobenius pairs of conjugate elements
cites work: Groups with finite Engel element
full work available at URL: s0001434621010314
OpenAlex ID: W3179039093
description: scientific article; zbMATH DE number 7369701