Date: Tue, 10 Oct 95 11:58:33 0100
From: T.Ward@uea.ac.uk

Hello.

If  G  is abelian, with finite torsion-free rank, and  A  is
an injective homomorphism from  G  to itself, does it
follow that the quotient  G/AG  is finite?

If so, is there a `formula' for the order? For example,
if  G  is  Z^d,  then the quotient group has order
given by the modulus of the determinant defining
A.

More suggestively, if  G  is  Q^d, (Q the rationals)
then the order (which is 1) is equal to the product
over all primes of the p-adic norm of the determinant of
the rational matrix defining A.

One would expect that there is a `maximum' size
associated with A: if  A  is a  k by k rational matrix,
and G_A is the group  Z^k[A;A^{-1}]   then for any
other  G  with rank  k  the quotient  G/AG  should be
smaller than  G_A/AG_A, which can be computed.

(This question arises from trying to understand a
problem in ergodic theory, which is my own area.
Apologies if the question is trivial!)

Tom Ward