Revista Matemática Iberoamericana


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Volume 30, Issue 2, 2014, pp. 537–550
DOI: 10.4171/RMI/792

Published online: 2014-07-08

Groups with restrictions on subgroups of infinite rank

Maria De Falco[1], Francesco de Giovanni[2], Carmela Musella[3] and Nadir Trabelsi[4]

(1) Università degli Studi di Napoli Federico II, Italy
(2) Università degli Studi di Napoli Federico II, Italy
(3) Università degli Studi di Napoli Federico II, Italy
(4) Ferhat Abbas University, Setif, Algeria

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 $G$ is a group of infinite rank such that all its proper subgroups of infinite rank have locally finite commutator subgroup, then the commutator subgroup $G'$ of $G$ is locally finite, provided that $G$ satisfies a suitable generalized solubility condition. Moreover, a similar result is obtained for groups whose proper subgroups of infinite rank are quasihamiltonian.

Keywords: Prüfer rank, strongly locally graded group, quasihamiltonian group

De Falco Maria, de Giovanni Francesco, Musella Carmela, Trabelsi Nadir: Groups with restrictions on subgroups of infinite rank. Rev. Mat. Iberoamericana 30 (2014), 537-550. doi: 10.4171/RMI/792