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 { G j : j ∈ J } where each G j is a subgroup of G satisfying the equality card ( G / G j ) = 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.