Groups with minimax commutator subgroup

Document Type: Research Paper

Authors

1 Dipartimento di Matematica e Applicazioni - University of Napoli "Federico II"

2 Dipartimento di Matematica e Applicazooni - University of Napoli "Federico II"

Abstract

A result of Dixon‎, ‎Evans and Smith shows that if $G$ is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian‎, ‎then $G$ itself has this property‎, ‎i.e‎. ‎the commutator subgroup of $G$ has finite rank‎. ‎It is proved here that if $G$ is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup‎, ‎then also the commutator subgroup $G'$ of $G$ is minimax‎. ‎A corresponding result is proved for groups in which the commutator subgroup of every proper subgroup has finite torsion-free rank.‎

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V. V. Belyaev and N. F. Sesekin (1975). On infinite groups of Miller-Moreno type. Acta Math. Acad. Sci. Hungar.. 26, 369-376
N. S. Cernikov (1990). Theorem on groups of finite special rank. Ukrain. Math. J.. 24, 855-861
M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi Groups with restrictions on subgroups of infinite rank. Rev. Mat. Iberoamericana, to appear.
M. R. Dixon, M. J. Evans and H. Smith (1999). Groups with all proper subgroups (finite rank)-by-nilpotent. Arch. Math. (Basel). 72, 321-327
M. R. Dixon and Z. Y. Karatas (2012). Groups with all subgroups permutable or of finite rank. Centr. Eur. J. Math.. 10, 950-957
S. Franciosi, F. de Giovanni and M. J. Tomkinson (1991). Groups with Cernikov conjugacy classes. J. Austral. Math. Soc. Ser. A. 50, 1-14
F. de Giovanni and M. Trombetti Groups whose proper subgroups of infinite rank have polycyclic conjugacy classes. Algebra Colloq., to appear.
J. Otal and J. M. Pena (1988). Minimal non-CC-groups. Comm. Algebra. 16, 1231-1242
Y. D. Polovickii (1964). Groups with extremal classes of conjugate elements. Sibirskii Mat. Z.. 5, 891-895
D. J. S. Robinson (1972). Finiteness conditions and generalized soluble groups. Springer, Berlin.
O. A. Yarovaya (2008). On groups whose all proper subgroups are close to abelian ones. Visn. Kyiv Univ. Ser. Fiz--Mat. Nauk. 4, 36-39