Geodesic
An extension of the Kegel–Wielandt theorem to locally finite groups
Glasgow mathematical journal, Tome 38 (1996) no. 2, pp. 171-176

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Franciosi, Silvana; Giovanni, Francesco de; Sysak, Yaroslav P. An extension of the Kegel–Wielandt theorem to locally finite groups. Glasgow mathematical journal, Tome 38 (1996) no. 2, pp. 171-176. doi: 10.1017/S0017089500031402
@article{10_1017_S0017089500031402,
     author = {Franciosi, Silvana and Giovanni, Francesco de and Sysak, Yaroslav P.},
     title = {An extension of the {Kegel{\textendash}Wielandt} theorem to locally finite groups},
     journal = {Glasgow mathematical journal},
     pages = {171--176},
     year = {1996},
     volume = {38},
     number = {2},
     doi = {10.1017/S0017089500031402},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031402/}
}
TY  - JOUR
AU  - Franciosi, Silvana
AU  - Giovanni, Francesco de
AU  - Sysak, Yaroslav P.
TI  - An extension of the Kegel–Wielandt theorem to locally finite groups
JO  - Glasgow mathematical journal
PY  - 1996
SP  - 171
EP  - 176
VL  - 38
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031402/
DO  - 10.1017/S0017089500031402
ID  - 10_1017_S0017089500031402
ER  - 
%0 Journal Article
%A Franciosi, Silvana
%A Giovanni, Francesco de
%A Sysak, Yaroslav P.
%T An extension of the Kegel–Wielandt theorem to locally finite groups
%J Glasgow mathematical journal
%D 1996
%P 171-176
%V 38
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031402/
%R 10.1017/S0017089500031402
%F 10_1017_S0017089500031402

[1] 1.Amberg, B., Franciosi, S. and de Giovanni, F., Products of groups, (Clarendon Press, Oxford, 1992). Google Scholar

[2] 2.Baer, R., Lokal endliche-auflösbare Gruppen mit endlichen Sylowuntergruppen, J. Reine Angew. Math, 239/240 (1969), 109–144. Google Scholar

[3] 3.Franciosi, S. and de Giovanni, F., On products of nilpotent groups, Ricerche Mat., to appear. Google Scholar

[4] 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. Google Scholar | DOI

[5] 5.Kazarin, L. S., On a problem of Szép, Math. USSR lzy. 28 (1987), 467–495. Google Scholar | DOI

[6] 6.Kegel, O. H., On the solvability of some factorized linear groups, Illinois J. Math. 9 (1965), 535–547. Google Scholar | DOI

[7] 7.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, (Springer, Berlin, 1972). Google Scholar | DOI

[8] 8.Sysak, Ya. P., Products of infinite groups, Akad. Nauk Ukrain. Inst. Mat. Kiev, Preprint 82.53 (1982). Google Scholar

Cité par Sources :