Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T06:16:36.295Z Has data issue: false hasContentIssue false
  • English
  • Français

An extension of the Kegel–Wielandt theorem to locally finite groups

Published online by Cambridge University Press:  18 May 2009

Silvana Franciosi ,
Francesco de Giovanni  and
Yaroslav P. Sysak
Silvana Franciosi
Affiliation:
Dipartimento Di Matematica E Applicazioni, Università Di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli (Italy)
Francesco de Giovanni
Affiliation:
Dipartimento Di Matematica E Applicazioni, Università Di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli (Italy)
Yaroslav P. Sysak
Affiliation:
Institute of Mathematics, Ukrainian Academy of Sciences, 252601 Kiev (Ukraine)
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 famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 2 , May 1996 , pp. 171 - 176
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

REFERENCES

1.Amberg, B., Franciosi, S. and de Giovanni, F., Products of groups, (Clarendon Press, Oxford, 1992).Google Scholar
2.Baer, R., Lokal endliche-auflösbare Gruppen mit endlichen Sylowuntergruppen, J. Reine Angew. Math, 239/240 (1969), 109–144.Google Scholar
3.Franciosi, S. and de Giovanni, F., On products of nilpotent groups, Ricerche Mat., to appear.Google Scholar
4.Franciosi, S., de Giovanni, F. and Sysak, Ya. P., On locally finite groups factorized by locally nilpotent subgroups, J. Pure Appl. Algebra, 106 (1996), 45–56.CrossRefGoogle Scholar
5.Kazarin, L. S., On a problem of Szép, Math. USSR lzy. 28 (1987), 467–495.CrossRefGoogle Scholar
6.Kegel, O. H., On the solvability of some factorized linear groups, Illinois J. Math. 9 (1965), 535–547.CrossRefGoogle Scholar
7.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, (Springer, Berlin, 1972).CrossRefGoogle Scholar
8.Sysak, Ya. P., Products of infinite groups, Akad. Nauk Ukrain. Inst. Mat. Kiev, Preprint 82.53 (1982).Google Scholar