Groups saturated with finite Frobenius groups
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.
scientific article; zbMATH DE number 7369701
scientific article