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Length functions on hypercentral groups

Published online by Cambridge University Press:  20 January 2009

David L. Wilkens
David L. Wilkens
Affiliation:
Department of Mathematics, University of Birmingham, Birmingham, B15 2TT
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In [6] the structure of any real valued length function on an abelian group G is determined. It is shown there, in Theorem 6.1., that such a length function is an extension of a non-Archimedean length function l1 on N by an Archimedean length function l2 on H=G/N. Any non-Archimedean length function is given by a chain of subgroups, as described in [5], and following from results of Nancy Harrison [2], the length l2 is essentially the absolute value function on a subgroup of R. In the situation above if N≠G then N is a subgroup of G whose elements have bounded lengths. In this paper we show that it is an easy consequence of techniques developed in [1] that this result can be extended to hypercentral groups, thus determining the structure of any length function in this case. We point out that the result does not extend to soluble groups. The infinite dihedral group D∞ is soluble. However if D∞ is regarded as a free product of two cyclic groups of order 2 and is given the length function associated with a free product, as described by Lyndon [3], then N is not a subgroup of D∞, and the lengths of its elements are unbounded.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 303 - 304
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

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