Topologically independent sets in precompact groups

In other words, we prove that if A is a subset of a precompact abelian group, then the topological subgroup generated by A is the Tychonoff direct sum of the topological cyclic subgroups 〈a〉, a∈A if and only if A is topologically independent. We show that precompactness can not be replaced by local compactness in this result.