Factoring groups into dense subsets

Let $G$ be a group of cardinality $\kappa>\aleph_0$ endowed with a topology $\tau$ such that $|U|=\kappa$ for every non-empty $U\in\tau$ and $\tau$ has a base of cardinality $\kappa$. We prove that $G$ could be factorized $G=AB$ (i.e. each $g\in G$ has unique representation $g=ab$, $a\in A$, $b\in B$) into dense subsets $A,B$, $|A|=|B|=\kappa$. We do not know if this statement holds for $\kappa = \aleph_0$ even if $G$ is a topological group.