Paper “On locally finite groups whose subgroups of infinite rank have some permutable property” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. Camp-Mora, M. R. Dixon, R. Ialenti, and F. Spagnuolo

On locally finite groups whose subgroups of infinite rank have some permutable property

Ann. Mat. Pura Appl. (4), 196(5):1855–1862, 2017

https://doi.org/10.1007/s10231-017-0642-7

Abstract

In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case.

2010 Mathematical Subject Classification: 20F19, 20F50

Keywords: Locally finite group, Section p-rank, Section rank, Special rank, Permutable, Sylow permutable, Semipermutable, S-semipermutable