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

  • Silvana Franciosi (a1), Francesco de Giovanni (a1) and Yaroslav P. Sysak (a2)

Extract

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.

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References

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1.Amberg, B., Franciosi, S. and de Giovanni, F., Products of groups, (Clarendon Press, Oxford, 1992).
2.Baer, R., Lokal endliche-auflösbare Gruppen mit endlichen Sylowuntergruppen, J. Reine Angew. Math, 239/240 (1969), 109144.
3.Franciosi, S. and de Giovanni, F., On products of nilpotent groups, Ricerche Mat., to appear.
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), 4556.
5.Kazarin, L. S., On a problem of Szép, Math. USSR lzy. 28 (1987), 467495.
6.Kegel, O. H., On the solvability of some factorized linear groups, Illinois J. Math. 9 (1965), 535547.
7.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, (Springer, Berlin, 1972).
8.Sysak, Ya. P., Products of infinite groups, Akad. Nauk Ukrain. Inst. Mat. Kiev, Preprint 82.53 (1982).

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