Recent efforts to generalize a classic result of Hajos  on the decomposition of finite abelian groups into direct sums of subsets (see Fuchs [1, Chap. XV]) led B. Gordon  to the following conjecture. If are r-dimensional row vectors over GF(3) such that: (i) Any weighted (±) sum of any even number of 's is nonzero, (ii) For each r-dimensional , there exists an s such that
Then there exists a subset of either 1 or 4 's which satisfies the same conditions.
This paper proves Gordon's conjecture.