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Residually finite groups of finite rank

Published online by Cambridge University Press:  24 October 2008

Alexander Lubotzky  and
Avinoam Mann
Alexander Lubotzky
Affiliation:
Hebrew University, Jerusalem
Avinoam Mann
Affiliation:
Hebrew University, Jerusalem
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Extract

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:

Theorem 1. A residually finite group of finite rank is virtually locally soluble.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 3 , November 1989 , pp. 385 - 388
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Huppert, B.. Endliche Gruppen, vol. 1 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[2]Lubotzky, A.. A group theoretic characterization of linear groups. J. Algebra 113 (1988), 207–214.CrossRefGoogle Scholar
[3]Lubotzky, A. and Mann, A.. Powerful p-groups, J. Algebra 105 (1987), 484–515.CrossRefGoogle Scholar
[4]Robinson, D. J. S.. Finiteness Conditions and Generalized Soluble Groups, Ergeb. Math. Grenzgeb. nos. 62, 63 (Springer-Verlag, 1972).Google Scholar
[5]Wehrfritz, B. A. F.. Infinite Linear Groups, Ergeb. Math. Grenzgeb. no. 76 (Springer-Verlag, 1967).Google Scholar