Iteration of operations on classes of groups
Let \({\mathcal K}\), \({\mathcal M}\), and \({\mathcal N}\) be the operations on classes of groups defined as follows: For every class of groups \({\mathfrak X}\), \({\mathcal K}{\mathfrak X}\) consists of all groups which are set-theoretical unions of their normal \({\mathfrak X}\)-subgroups, \({\mathcal M}{\mathfrak X}\) consists of all groups in which every finite set of elements lies in a normal \({\mathfrak X}\)-subgroup and \({\mathcal N}{\mathfrak X}\) consists of all groups generated by their normal \({\mathfrak X}\)-subgroups. For the class \({\mathfrak A}\) of abelian groups and every ordinal \(\alpha\), \textit{P. G. Kontorovich} [Mat. Sb., Nov. Ser. 28(70), 79-88 (1951; Zbl 0042.017)] constructed the class \({\mathcal K}^{\alpha}{\mathfrak A}\) by iterated application of \({\mathcal K}\) and introduced the closure operations generated by \({\mathcal K}\) as \(\bar {\mathcal K}{\mathfrak A}=\cup_{\alpha}{\mathcal K}^{\alpha}{\mathfrak A}.\) Then in his paper [Algebra Mat. Logika, 62-71 (1966; Zbl 0204.346)] \textit{L. A. Kaluzhnin} constructed the class \(\bar {\mathcal M}{\mathfrak F}\), where \({\mathfrak F}\) is the class of finite groups and in that paper he posed the question whether there exist groups in \(\bar {\mathcal M}{\mathfrak F}\) not belonging to \({\mathcal M}^{\omega +1}{\mathfrak F}\). The author provides in this paper an affirmative answer to this question. His study is based on comparing the actions of \({\mathcal M}\), \({\mathcal K}\) and \({\mathcal N}\) and he proves that \(\sigma({\mathfrak A},{\mathcal K})\leq \omega^{\omega}+1,\quad \sigma({\mathfrak F},{\mathcal M})\leq \omega^{\omega}+1\) where \(\sigma\) (\({\mathfrak A},{\mathcal K})\) denotes the minimal ordinal satisfying \({\mathcal K}^{\sigma}{\mathfrak A}=\bar {\mathcal K}{\mathfrak A}.\) Using wreath products of groups, he shows that \({\mathcal M}^{\omega^ 2}{\mathfrak X}\neq {\mathcal M}^{\omega^ 2+1}{\mathfrak X}\) and \({\mathcal N}^{\omega}{\mathfrak X}\neq {\mathcal N}^{\omega +1}{\mathfrak X}\) for some classes of groups \({\mathfrak X}\). In particular, he shows that \(\sigma({\mathfrak F},{\mathcal M})\geq \omega^ 2+1.\)
scientific article; zbMATH DE number 3851360