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On Elliptically Embedded Subgroups of Soluble Groups

Published online by Cambridge University Press:  20 November 2018

A. H. Rhemtulla  and
J. S. Wilson
A. H. Rhemtulla
Affiliation:
University of Alberta, Edmonton, Alberta
J. S. Wilson
Affiliation:
University of Alberta, Edmonton, Alberta
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We call a subset X of a group an elliptic set if there is an integer n such that each element of the group generated by X can be written as a product of at most n elements of X ∪ X−1. The terminology is due to Philip Hall, who investigated elliptic sets in lectures given in Cambridge in the 1960's. Hall was chiefly interested in sets X which are unions of conjugacy classes, but among other things he proved that if H, K are subgroups of a finitely generated nilpotent group then their union H ∪ K is elliptic. We shall say that a subgroup H of an arbitrary group G is elliptically embedded in G, and we write H ee G, if H ∪ K is an elliptic set for each subgroup K of G. Thus H ee G if for each subgroup K there is an integer n (depending on K) such that

where the product has 2n factors.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 956 - 968
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Hecke, E., Vorlesungen über die Théorie der algebraischen Zahlen, 2nd Edition (Akademische Verlagsgesellschaft, Geest U. Portig K.-G., Leipzig, 1954).Google Scholar
2. Meier, D. and Rhemtulla, A. H., On torsion-free groups of finite rank, Can. J. Math. 6 (1984), 1067–1080.Google Scholar
3. Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Parts I and II (Springer-Verlag, New York, 1972).Google Scholar