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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Real length functions in groups

Author: Nancy Harrison
Journal: Trans. Amer. Math. Soc. 174 (1972), 77-106
MSC: Primary 20F99
MathSciNet review: 0308283
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Abstract: This paper is a study of the structure of a group G equipped with a 'length' function from G to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon's work on groups with integer-valued functions. A real length function is a function which assigns to each $ g \in G$ a nonnegative real number $ \vert g\vert$ such that the following axioms are satisfied:

\begin{displaymath}\begin{array}{*{20}{c}} {{{\text{A}}_0}.\quad \vert x\vert < ... ... \geq m\;{\text{imply}}\;c(x,z) \geq m.} \hfill \\ \end{array} \end{displaymath}

In this paper structure theorems are obtained for the cases when G is abelian and when G can be generated by two elements. We first prove that if G is abelian, then G is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of G. We prove that if G can be generated by two elements, then G is either free or abelian.

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Keywords: Nielsen transformation, groups with functions to reals
Article copyright: © Copyright 1972 American Mathematical Society

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