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Residually finite groups of finite rank

  • Alexander Lubotzky (a1) and Avinoam Mann (a1)

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.

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[1]B. Huppert . Endliche Gruppen, vol. 1 (Springer-Verlag, 1967).

[2]A. Lubotzky . A group theoretic characterization of linear groups. J. Algebra 113 (1988), 207214.

[3]A. Lubotzky and A. Mann . Powerful p-groups, J. Algebra 105 (1987), 484515.

[4]D. J. S. Robinson . Finiteness Conditions and Generalized Soluble Groups, Ergeb. Math. Grenzgeb. nos. 62, 63 (Springer-Verlag, 1972).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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