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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)
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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

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