It is known that a (generalized) soluble group whose proper subgroups of infinite rank are abelian either is abelian or has finite rank. It is proved here that if is a group of infinite rank such that all its proper subgroups of infinite rank have locally finite commutator subgroup, then the commutator subgroup of is locally finite, provided that satisfies a suitable generalized solubility condition. Moreover, a similar result is obtained for groups whose proper subgroups of infinite rank are quasihamiltonian.
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Maria De Falco, Francesco de Giovanni, Carmela Musella, Nadir Trabelsi, Groups with restrictions on subgroups of infinite rank. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 537–550DOI 10.4171/RMI/792