Infinite groups with many generalized normal subgroups. (Q2837244)

This paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal. The main results of the article are the following theorems.NEWLINENEWLINE Let \(G\) be a locally finite group with finitely many normalizers of infinite subgroups which are neither almost normal nor nearly normal. Then either the commutator subgroup of \(G\) is finite or \(G\) is a Chernikov group whose infinite subgroups are almost normal.NEWLINENEWLINE Let \(G\) be a non-periodic group in which every infinite subgroup is either almost normal or nearly normal. Then either the commutator subgroup of \(G\) is finite or all infinite subgroups of \(G\) are almost normal.NEWLINENEWLINE Let \(G\) be a non-periodic group in which the elements of finite order form a subgroup. If \(G\) has finitely many normalizers of infinite subgroups which are neither almost normal nor nearly normal, then either all infinite subgroups of \(G\) are almost normal or all infinite subgroups of \(G\) are nearly normal.