- Published 2010

Alexandroff and Hopf1 offer a proof of the following theorem.2 If U is a sub-group of an Abelian group J and m is an integer such that m = 0 or m ja 2, then rm(J)^rm(U)+rm(J—U). The proof is incorrect and the following example shows that the theorem is, in fact, not true. Example 1. Let / be the group of integers mod 4, and U the subgroup generated by 2; r2(J) = 1, r2(U) = 1, r2(J— U) = 1. The proof referred to is correct if m = 0, and the authors, in fact, prove that ro(J) =ro(U)+r0(J— U). In what follows we shall assume this, and that all groups considered are finitely generated and Abelian.3

@inproceedings{HAMSTROM2010LinearII,
title={Linear Independence in Abelian Groups},
author={MARY-ELIZABETH HAMSTROM},
year={2010}
}