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A permutability problem in infinite groups and Ramsey's theorem

Published online by Cambridge University Press:  17 April 2009

Alireza Abdollahi
Affiliation:
Department of Mathematics, University of Isfahan, Isfahan 81744, Iran and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran e-mail: [email protected]@math.ui.ac.ir
Aliakbar Mohammadi Hassanabadi
Affiliation:
Department of Mathematics, University of Isfahan, Isfahan 81744, Iran and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran e-mail: [email protected]@math.ui.ac.ir
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Abstract

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We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which ever infinite subset contains n mutually disjoint non-empty subsets X1,…,Xn such that X1XnXσ(1)xσ(n) ≠ θ for some non-identity permutation σ on the set {1,…,n}.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Abdollahi, A. and Hassanabadi, A. Mohammadi, ‘A characterization of infinite abelian groups’, Bull. Iranian Math. Soc. 24 (1998), 4147.Google Scholar
[2]Abdollahi, A., Hassanabadi, A. Mohammadi and Taeri, B., ‘A property equivalent to n-permutability for infinite groups’, J. Algebra 221 (1999), 570578.CrossRefGoogle Scholar
[3]Abdoilahi, A., Hassanabadi, A. Mohammadi and Taeri, B., ‘An n-rewritability criterion for infinite groups’, Comm. Algebra (to appear).Google Scholar
[4]Babai, L. and Sós, T.S., ‘Sidon sets in groups and induced subgraphs of Cayley groups’, European J. Combin. 6 (1985), 101114.CrossRefGoogle Scholar
[5]Blyth, R.D., ‘Rewriting products of group elements I’, J. Algebra 116 (1988), 506521.CrossRefGoogle Scholar
[6]Blyth, R.D., ‘Rewriting products of group elements II’, J. Algebra 118 (1988), 249259.Google Scholar
[7]Curzio, M., Longobardi, P. and Maj, M., ‘On a combinatorial problem in group theory’, (in Italian), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 74 (1983), 136142.Google Scholar
[8]Curzio, M., Longobardi, P., Maj, M. and Rhemtulla, A., ‘Groups with many rewritable products’, Proc. Amer. Math. Soc. 115 (1992), 931934.Google Scholar
[9]curzio, M., Longobardi, P., Maj, M. and Robinson, D.J.S., ‘A permutational property of groups’, Arch. Math. (Basel) 44 (1985), 385389.CrossRefGoogle Scholar
[10]Groves, J.R.J., ‘A conjecture of Lennox and Wiegold concerning supersoluble groups’, J. Austral. Math. Soc. Ser. 35 (1983), 218220.CrossRefGoogle Scholar
[11]Kargapolov, M.I. and Merzljakov, Ju.I., Fundamentals of the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[12]Lennox, J.C. and Wiegold, J., ‘Extensions of a problem of Paul Erdös on groupsJ. Austral. Math. Soc. Ser. A 31 (1981), 459463.CrossRefGoogle Scholar
[13]Neumann, B.H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 21 (1976), 467472.CrossRefGoogle Scholar
[14]Ramsey, F.P., ‘On a problem of formal logic’, Proc. London Math. Soc. (2) 30 (1929). 264286.Google Scholar