GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN. Issue 1 (8th August 2020) - ESPOSITO, DARIO; DE GIOVANNI, FRANCESCO; TROMBETTI, MARCO
Abstract : If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1}, \mathfrak{X}_{2}, …, \mathfrak{X}_{k}, …$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$ . In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$ -groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$ . Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$ .