Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T21:02:33.430Z Has data issue: false hasContentIssue false
  • English
  • Français

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: mdefalco@unina.it, degiovan@unina.it, cmusella@unina.it
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: mdefalco@unina.it, degiovan@unina.it, cmusella@unina.it
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: mdefalco@unina.it, degiovan@unina.it, cmusella@unina.it
Y. P. Sysak
Affiliation:
Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchenkivska 3, 01601 Kiev, Ukraine, e-mail: sysak @imath.kiev.ua
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Buckley, J. T., Lennox, J. C., Neumann, B. H., Smith, H. and Wiegold, J., ‘Groups with all subgroups normal-by-finite’, J. Aust. Math. Soc. Ser. A 59 (1995), 384–398.CrossRefGoogle Scholar
[2]de Giovanni, F., Musella, C. and Sysak, Y. P., ‘Groups with almost modular subgroup lattice’, J. Algebra 243 (2001), 738–764.CrossRefGoogle Scholar
[3]Dixon, M. R., Sylow theory, formations and fitting classes in locally finite groups, Series in Algebra 2 (World Scientific, Singapore, 1994).CrossRefGoogle Scholar
[4]De Falco, M., de Giovanni, F. and Musella, C., ‘Groups in which every subgroup is permutable-byfinite’, Comm. Algebra 32 (2004), 1007–1017.CrossRefGoogle Scholar
[5]De Falco, M., de Giovanni, F., Musella, C. and Sysak, Y. P., ‘Groups in which every subgroup is nearly permutable’. Forum Math. 15 (2003), 665–677.CrossRefGoogle Scholar
[6]Iwasawa, K., ‘Über die endlichen Gruppen und die Verbände ihrer Untergruppen’, J. Fac. Sci. lmp. Univ. Tokyo Sect. 14 (1941), 171–199.Google Scholar
[7]Iwasawa, K., ‘On the structure of infinite M-groups’, Japan J. Math. 18 (1943), 709–728.CrossRefGoogle Scholar
[8]Möhres, W., ‘Hyperzentrale Torsiongruppen deren Untergruppen alle subnormal sind’, Illinois J. Math. 35 (1991), 147–157.CrossRefGoogle Scholar
[9]Neumann, B. H., ‘Groups with finite classes of conjugate subgroups’, Math. Z. 63 (1955), 76–96.CrossRefGoogle Scholar
[10]Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, Berlin, 1972).CrossRefGoogle Scholar
[11]Schmidt, R., Subgroup lattices of groups (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
[12]Stonehewer, S. E., ‘Permutable subgroups of infinite groups’, Math. Z. 125 (1972), 1–16.CrossRefGoogle Scholar