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On polygonal products of finitely generated abelian groups

Published online by Cambridge University Press:  17 April 2009

Goansu Kim
Goansu Kim
Affiliation:
Department of MathematicsKangnung National UniversityKangnung, Kangwon-Do, 210-702Korea
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Abstract

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We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 3 , June 1992 , pp. 453 - 462
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Allenby, R.B.J.T. and Gregorac, R.J., ‘On locally extended residually finite groups’, in Lecture Notes in Mathematics 319, pp. 9–17 (Springer-Verlag, New York, 1973).Google Scholar
[2]Allenby, R.B.J.T. and Tang, C.Y., ‘The residual finiteness of some one-relator groups with torsion’, J. Algebra 71 (1981), 132–140.CrossRefGoogle Scholar
[3]Allenby, R.B.J.T. and Tang, C.Y., ‘On the residual finiteness of certain polygonal products’, Canad. Math. Bull. 32 (1989), 11–17.CrossRefGoogle Scholar
[4]Baumslag, G., ‘On the residual finiteness of generalized free products of nilpotent groups’, Trans. Amer. Math. Soc. 106 (1963), 193–209.CrossRefGoogle Scholar
[5]Brunner, A.M., Frame, M.L., Lee, Y.W. and Wielenberg, N.J., ‘Classifying the torsion-free subgroups of the Picard group’, Trans. Amer. Math. Soc. 282 (1984), 205–235.CrossRefGoogle Scholar
[6]Karrass, A., Pietrowski, A. and Solitar, D., ‘The subgroups of polygonal products of groups’, (Unpublished manuscript).Google Scholar
[7]Kim, G., ‘Conjugacy and subgroup separability of generalized free product’, (Ph.D. thesis submitted to University of Waterloo, 1991).Google Scholar
[8]Kim, G., ‘Cyclic subgroup separability of generalized free products’, (Manuscript 1991).Google Scholar
[9]Kim, G. and Tang, C.Y., ‘On the residual finiteness of polygonal products of nilpotent groups’, Canad. Math. Bull. (to appear).Google Scholar
[10]Magnus, W., Karrass, A. and Solitar, D., ‘Combinatorial Group Theory’, in Pure and Applied Math. XIII (Wiley-Interscience, New York, London, Sydney, 1966).Google Scholar
[11]Wehrfritz, B.A. F., ‘The residual finiteness of some generalized free products’, J. London Math. Soc. 24 (1981), 123–126.CrossRefGoogle Scholar