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A Finite Index Property of Certain Solvable Groups

Published online by Cambridge University Press:  20 November 2018

A. H. Rhemtulla  and
H. Smith
A. H. Rhemtulla
Affiliation:
University of Alberta, Edmonton, Canada T6G 2G1
H. Smith
Affiliation:
York University, Downsview, Canada M3J 1P3
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Abstract

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A group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

Keywords

20E3420F16
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 485 - 489
Copyright
Copyright © Canadian Mathematical Society 1984

Footnotes

1

Research partially supported by a grant from Natural Sciences and Engineering Research Council of Canada.

References

1. Meier, D. and Rhemtulla, A. H., On Torsion-Free Groups of Finite Rank. Can. J. Math, (to appear).Google Scholar
2. Passman, D. S., The Algebraic Structure of Group Rings. A Wiley-Interscience Publication, John Wiley & Sons, New York 1977.Google Scholar
3. Rhemtulla, A. H. and Wehrfritz, B. A. F., Isolators in Soluble Groups of Finite Rank. Rock Mountain J. Math. 14 (1984), 415-421.Google Scholar
4. Rhemtulla, A. H., Weiss, A. and Yousif, M., Solvable Groups with ir-isolators. Proc. Amer. Math. Soc. 90 (1984), 173-177.Google Scholar
5. Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups, Parts 1 and 2. Springer Verlag 1972.CrossRefGoogle Scholar
6. Robinson, D. J. S., A Course in the Theory of Groups. Graduate text in Mathematics vol. 80 Springer Verlag 1980.Google Scholar
7. Smith, H., On Certain Subgroups of a Join of Subnormal Subgroups. Glasgow J. Math. 25 (1984), 103-105.CrossRefGoogle Scholar