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Ascendant subgroups of factorized FC-groups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  26 February 2010

Falih A. M. Aldosray
Falih A. M. Aldosray
Affiliation:
Department of Mathematics, Umm Al-Qura University, Makkah, P.O. Box 7296, Saudi Arabia.
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Extract

Wielandt [4] has shown that a common subnormal subgroup of two permutable subgroups of finite group is subnormal in their product. When G is infinite it seems unlikely that Wielandt's theorem will still be true, but an example illustrating this appears to be difficult, even if G is an FC-group (that is groups in which each element has only finitely many conjugates; see [2]). However, if we replace subnormality by ascendancy we have the following.

MSC classification

Secondary: 20E07: Subgroup theorems; subgroup growth
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 188 - 189
Copyright
Copyright © University College London 1995

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References

1. Lennox, J. C. and Stonehewer, S. E.. Subnormal subgroups of groups (Clarendon Press, Oxford, 1987).Google Scholar
2. Tomkinson, M. J.. FC-groups. Research note in mathematics, 96 (Pitman, London, 1984).Google Scholar
3. Wielandt, H.. Uber das Erzeugnis Paarweise Kosubnormaler Untergruppen. Arch. Math. (Basel), 35 (1980), 1–7.CrossRefGoogle Scholar
4. Wielandt, H.. Subnormalitat in Faktorisierten endlichen Gruppen. J. Algebra, 69 (1981), 305–311.CrossRefGoogle Scholar