On minimal non- $$M_rC$$ -groups

Abstract

A group \(G\) is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let \(D(G)\) denotes the subgroup of \(G\) generated by all the Chernikov divisible normal subgroups of \(G\). If \(G\) is a soluble-by-finite minimax group and if \(D(G)=1\) , then \(G\) is said to be a reduced minimax group. Also \(G\) is said to be an \( M_{r}C\)-group (respectively, \(PC\)-group), if \(G/C_{G} \left(x^{G}\right)\) is a reduced minimax (respectively, polycyclic-by-finite) group for all \(x\in G\) . These are generalisations of the familiar property of being an \(FC\)-group. Finally, if \(\mathfrak X \) is a class of groups, then \(G\) is said to be a minimal non-\(\mathfrak X \)-group if it is not an \(\mathfrak X \)-group but all of whose proper subgroups are \(\mathfrak X \)-groups. Belyaev and Sesekin characterized minimal non-\(FC\)-groups when they have a non-trivial finite or abelian factor group. Here we prove that if \(G\) is a group that has a proper subgroup of finite index, then \(G\) is a minimal non-\(M_{r}C\)-group (respectively, non-\(PC\)-group) if, and only if, \(G\) is a minimal non-\(FC\)-group.