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Embedding theorems for groups with an integer-valued length function

Published online by Cambridge University Press:  24 October 2008

I. M. Chiswell
I. M. Chiswell
Affiliation:
Queen Mary College, University of London
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(1) In this paper we are concerned with normalized integer-valued length functions on a group G, that is, mappings l:G→ℤ satisfying three axioms:

(A1′)l(1) = 0

(A2) l(x) = l(x−1)for all x∈G

(A4) d(x, y) > d(x, z) implies that d(x, z) = d(y, z) for all x, y and z in G, where d(x, y) = ½(l(x) + l(y)−l(xy−1)).

These axioms were first considered by Lyndon(6), where their significance is discussed. Lyndon's axiom A1, which we shall not use, stated that l(x) = 0 if and only if x = 1. His axiom A 3 was that d(x, y) ≥ 0 for all x and y in G, but it was noted in (2) that this follows from A1′, A2 and A4. In particular, taking x = y, we find that l(x) ≥ 0 for all x in G. The other major axioms used in (6) were:

(A0) l(x2) > l(x), provided that x ≠ 1, and (A5) d(x, y) + d(xl, y1) > l(x) = l(y) implies that x = y.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 3 , May 1979 , pp. 417 - 429
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

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