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Permutable word products in groups

Published online by Cambridge University Press:  17 April 2009

P.S. Kim  and
A.H. Rhemtulla
P.S. Kim
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
A.H. Rhemtulla
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
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Abstract

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Let u(x1,…,xn) = x11 … x1m be a word in the alphabet x1, …,xn such that x1i ≠ x1i for all i = 1,…, m − 1. If (H1, …, Hn) is an n-tuple of subgroups of a group G then denote by u(H1, …, Hn) the set {u(h1,…,hn) | hi ∈ Hi}. If σ ∈ Sn then denote by uσ(H1,…,Hn) the set u(Hσ(1),…,Hσ(n)). We study groups G with the property that for each n-tuple (H1,…,Hn) of subgroups of G, there is some σ ∈ Sn σ ≠ 1 such that u(H1,…,Hn) = uσ(H1,…,Hn). If G is a finitely generated soluble group then G has this property for some word u if and only if G is nilpotent-by-finite. In the paper we also look at some specific words u and study the properties of the associated groups.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 2 , October 1989 , pp. 243 - 254
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Blyth, R.D., ‘Rewriting products of group elements I’, J. Algebra 116 (1988), 506–521.Google Scholar
[2]Curzio, M., Longobardi, P., Maj, M. and Robinson, D.J.S., ‘A permutational property of groups’, Arch. Math. (Basel) 44 (1985), 385–389.Google Scholar
[3]Evertse, J.H., ‘On sums of S-units and linear recurrences’, Compositio Math. 53 (1984), 225–244.Google Scholar
[4]Kaplansky, I., ‘Groups with representations of bounded degree’, Canad. J. Math. 1 (1949), 105–112.Google Scholar
[5]Longobardi, P., Maj, M. and Rhemtulla, A.H., ‘Periodic groups with, permutable subgroup products’, Math. Proc. Cambridge Philos. Soc. (to appear).Google Scholar
[6]Rhemtulla, A.H. and Weiss, A.R., ‘Groups with, permutable subgroup products’, in Group Theory: Proceedings of the 1987 Singapore Conference, Ed. Cheng, K.N. and Leong, Y.K., pp. 485–495 (DeGruyter 1989).CrossRefGoogle Scholar
[7]Rhemtulla, A.H. and Wilson, J.S., ‘On elliptically embedded subgroups of soluble groups’, Canad. J. Math. 39 (1987), 956–968.Google Scholar
[8]Smith, H., ‘Elliptically embedded subgroups and nilpotent products’. (submitted).Google Scholar
[9]Van der Poorten, A.J., ‘Additive relations in number fields’, Séminaire de Théorie des Nombres de Paris, (1982–1983), 259–266.Google Scholar