Groups whose non-normal subgroups have small commutator subgroup

Abstract

The structure of groups whose non-normal subgroups have a finite commutator subgroup is investigated. In particular, it is proved that if \(k\) is a positive integer and \(G\) is a locally graded group in which every non-normal subgroup has finite commutator subgroup of order at most \(k\), then the commutator subgroup of \(G\) is finite. Moreover, groups with finitely many normalizers of subgroups with large commutator subgroup are studied.