On thick subgroups of uncountable σ-compact locally compact commutative groups

We show that, for any uncountable commutative group (G,+), there exists a countable covering where each Gj is a subgroup of G satisfying the equality card(G/Gj)=card(G). This purely algebraic fact is used in certain constructions of thick and nonmeasurable subgroups of an uncountable σ-compact locally compact commutative group equipped with the completion of its Haar measure.