Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-08T23:46:07.226Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups in which every subgroup is modular-by-finite

Published online by Cambridge University Press:  17 April 2009

M. De Falco ,
F. De Giovanni  and
C. Musella
M. De Falco
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico0 II”, Complesso Universitario Monte S. Angelo, Via Cintia I 80126, Napoli, Italy
F. De Giovanni
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico0 II”, Complesso Universitario Monte S. Angelo, Via Cintia I 80126, Napoli, Italy
C. Musella
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico0 II”, Complesso Universitario Monte S. Angelo, Via Cintia I 80126, Napoli, Italy
Rights & Permissions [Opens in a new window]

Extract

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 group G is called a BCF-group if there is a positive integer κ such that |X : XG| ≤ κ for each subgroup X of G. The structure of BCF-groups has been studied by Buckley, Lennox, Neumann, Smith and Wiegold; they proved in particular that locally finite groups with the property BCF are Abelian-by-finite. As a group lattice version of this concept, we say that a group G is a BMF-group if there is a positive integer κ such that every subgroup X of G contains a modular subgroup Y of G for which the index |X : Y| is finite and the number of its prime divisors with multiplicity is bounded by κ (it is known that that such number can be characterised by purely lattice-theoretic considerations, and so it is invariant under lattice isomorphisms of groups). It is proved here that any locally finite BMF-group contains a subgroup of finite index with modular subgroup lattice.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 3 , June 2004 , pp. 441 - 450
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Ashmanov, I.S. - Ol'shanskii, A.Y., ‘Abelian and central extensions of aspherical groups’, Soviet. Math. (Iz. VUZ) 29 (1985), 65–82.Google Scholar
[2]Buckley, J.T., Lennox, J.C., Neumann, B.H., Smith, H. and Wiegold, J., ‘Groups with all subgroups normal-by-finite’, J. Austral. Math. Soc. Ser. A 59 (1995), 384–398.CrossRefGoogle Scholar
[3]De Falco, M., de Giovanni, F. and Musella, C., ‘Groups in which every subgroup is permutable-by-finite’, Comm. Algebra 32 (2004), 1007–1017.CrossRefGoogle Scholar
[4]De Falco, M., de Giovanni, F., Musella, C. and Schmidt, R., ‘Detecting the index of a subgroup in the subgroup lattice’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[5]De Falco, M., de Giovanni, F., Musella, C. and Sysak, Y.P., ‘Periodic groups with nearly modular subgroup lattice’, Illinois J. Math. 47 (2003), 189–205.CrossRefGoogle Scholar
[6]Dixon, M.R., Sylow theory, formations and Fitting classes in locally finite groups (World Scientific, Singapore, 1994).CrossRefGoogle Scholar
[7]Franciosi, S., de Giovanni, F. and Sysak, Y.P., ‘Groups with many polycyclic-by-nilpotent subgroups’, Ricerche Mat. 48 (1999), 361–378.Google Scholar
[8]de Giovanni, F. and Musella, C., ‘Groups with nearly modular subgroup lattice’, Colloq. Math. 88 (2001), 13–20.CrossRefGoogle Scholar
[9]de Giovanni, F., Musella, C. and Sysak, Y.P., ‘Groups with almost modular subgroup lattice’, J. Algebra 243 (2001), 738–764.CrossRefGoogle Scholar
[10]Iwasawa, K., ‘Über die endlichen Gruppen und die Verbände ihrer Untergruppen’, J. Fac. Sci. Imp. Univ. Tokyo 4 (1941), 171–199.Google Scholar
[11]Iwasawa, K., ‘On the structure of infinite M-groups’, Japan J. Math. 18 (1943), 709–728.CrossRefGoogle Scholar
[12]Neumann, B.H., ‘Groups with finite classes of conjugate subgroups’, Math. Z. 63 (1955), 76–96.CrossRefGoogle Scholar
[13]Robinson, D.J.S., Finiteness conditions and generalized soluble groups (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
[14]Schmidt, R., ‘Gruppen mit modularem Untergruppenverband’, Arch. Math. (Basel) 46 (1986), 118–124.CrossRefGoogle Scholar
[15]Schmidt, R., Subgroup lattices of groups (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
[16]Smith, H. and Wiegold, J., ‘Locally graded groups with all subgroups normal-by-finite’, J. Austral. Math. Soc. Ser. A 60 (1996), 222–227.CrossRefGoogle Scholar
[17]Stonehewer, S.E., ‘Permutable subgroups of infinite groups’, Math. Z. 125 (1972), 1–16.CrossRefGoogle Scholar
[18]Stonehewer, S.E., ‘Modular subgroups of infinite groups’, Sympos. Math. 17 (1976), 207–214.Google Scholar
[19]Stonehewer, S.E., ‘Modular subgroup structure in infinite groups’, Proc. London Math. Soc. (3) 32 (1976), 63–100.CrossRefGoogle Scholar