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