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The question of determining
which finite metabelian groups may be the homomorphic image of a given knot group
G is considered in this paper. As a starting point, it is shown that a homomorphism
of a knot group onto a metabelian group H such that [H : H′] = n must factor
through Zn An, where An is the homology group of the n-fold cyclic covering
space.
This is similar to a theorem of Burde [1 Satz 4|, and Reyner [5] has also proven a
similar result, showing in effect that such a homomorphism must factor
through Z ⊘ An. Now, An can be given the structure of a module over the
ring Z⟨t⟩ of L-polynomials, and the problem of determining the metabelian
factor groups of G can be reduced to determining the factor modules of
An.
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