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Commensurability and elementary equivalence of polycyclic groups

Published online by Cambridge University Press:  17 April 2009

Deborah Raphael
Deborah Raphael
Affiliation:
Universidade de Sāo PauloInstituto de Matemática e EstatísticaCaixa Postal 66281 – CEP 05389–970Sāo Paulo – Brasil
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Abstract

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It is shown that two polycyclic-by-finite groups G and H, satisfying the same sentences with one alternation of quantifiers, are commensurable. In fact we show something stronger: given n > 1 there is a subgroup Hn of H and a subgroup Gn of G such that Hn≃G, Gn≃H and the indices [G: Gn] and [H: Hn] are finite and prime to n.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 3 , June 1996 , pp. 425 - 439
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Baumslag, G., Lecture notes on nilpotent groups, Regional Conference Series in Math., 2 (American Mathematical Society, Providence, RI, 1971).Google Scholar
[2]Chang, C.C. and Keisler, H.J., Model theory (North-Holland, Amsterdam, New York, 1973).Google Scholar
[3]Grunewald, F.J. and Scharlau, R., ‘A note on finitely generated torsion-free nilpotent groups of class 2’, J. Algebra 58 (1979), 162–175.CrossRefGoogle Scholar
[4]Lang, S., Algebraic number theory, Graduate Texts in Math 110 (Springer Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar
[5]Oger, F., ‘Elementary equivalence and genus of finitely generated nilpotent groups’, Bull. Austral. Math. Soc. 37 (1988), 61–68.CrossRefGoogle Scholar
[6]Oger, F., ‘Équivalence élémentaire entre groupes finis-par-abéliens de type fini’, Comment. Math. Helv. 57 (1982), 469–480.CrossRefGoogle Scholar
[7]Oger, F., ‘Cancellation and elementary equivalence of finitely generated finite-by-nilpotent groups’, J. London Math. Soc. 44 (1990), 173–183.Google Scholar
[8]Oger, F., ‘Cancellation and elementary equivalence of groups’, J. Pure Appl. Algebra 30 (1983), 293–299.CrossRefGoogle Scholar
[9]Oger, F., ‘Elementary equivalence and profinite completions: a characterization of finitely generated abelian-by-finite groups’, Proc. Amer. Math. Soc. 103 (1988), 1041–1048.CrossRefGoogle Scholar
[10]Rhemtulla, A.H., ‘Commutators of certain finitely generated soluble groups’, Canad. J. Math. 21 (1969), 1160–1164.CrossRefGoogle Scholar
[11]Romankov, V.A., ‘Width of verbal subgroups in solvable groups’, Algebra and Logic 21 (1982), 41–49.CrossRefGoogle Scholar
[12]Segal, D., Polycyclic groups (Cambridge University Press, Cambridge, 1983).CrossRefGoogle Scholar