Revista Matemática Iberoamericana

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Volume 24, Issue 3, 2008, pp. 745–764
DOI: 10.4171/RMI/555

Published online: 2008-12-31

Infinite groups with many permutable subgroups

Adolfo Ballester-Bolinches[1], L. A. Kurdachenko[2], J. Otal[3] and T. Pedraza[4]

(1) Universitat de València, Burjassot (Valencia), Spain
(2) National Dnepropetrovsk University, Ukraine
(3) Universidad de Zaragoza, Spain
(4) Universidad Politécnica de Valencia, Spain

A subgroup $H$ of a group $G$ is said to be \textit{permutable in $G$}, if $HK = KH$ for every subgroup $K$ of $G$. A result due to Stonehewer asserts that every permutable subgroup is ascendant although the converse is false. In this paper we study some infinite groups whose ascendant subgroups are permutable ($AP$--groups). We show that the structure of radical hyperfinite $AP$--groups behave as that of finite soluble groups in which the relation \textit{to be a permutable subgroup} is transitive ($PT$--groups).

Keywords: radical groups, hyper--$\mathfrak{X}$--groups, $AP$--groups, $PT$--groups

Ballester-Bolinches Adolfo, Kurdachenko L., Otal J., Pedraza T.: Infinite groups with many permutable subgroups. Rev. Mat. Iberoamericana 24 (2008), 745-764. doi: 10.4171/RMI/555