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The structure of groups whose subgroups are permutable-by-finite

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

M. De Falco ,
F. De Giovanni ,
C. Musella  and
Y. P. Sysak
M. De Falco
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy, e-mail: [email protected], [email protected], [email protected]
F. De Giovanni
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy, e-mail: [email protected], [email protected], [email protected]
C. Musella
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy, e-mail: [email protected], [email protected], [email protected]
Y. P. Sysak
Affiliation:
Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchenkivska 3, 01601 Kiev, Ukraine, e-mail: sysak @imath.kiev.ua
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Abstract

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A subgroup H of a group G is said to be permutable if HX = XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a Q F-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite Q F-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if G is a locally finite group such that H/HG is finite for every subgroup H, then G contains an abelian subgroup of finite index.

MSC classification

Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 1 , August 2006 , pp. 35 - 48
Copyright
Copyright © Australian Mathematical Society 2006

References

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