  Bilinear Bogolyubov argument states that if we start with a dense set $A
\subseteq \mathbb{F}_p^n \times \mathbb{F}_p^n$ and carry out sufficiently many
steps where we replace every row or every column of $A$ by the set difference
of it with itself, then inside the resulting set we obtain a bilinear variety
of codimension bounded in terms of density of $A$. In this paper, we generalize
the bilinear Bogolyubov argument to arbitrary finite abelian groups. Namely, if
$G$ and $H$ are finite abelian groups and $A \subseteq G \times H$ is a subset
of density $\delta$, then the procedure above applied to $A$ results in a set
that contains a bilinear analogue of a Bohr set, with the appropriately defined
codimension bounded above by $\log^{O(1)} (O(\delta^{-1}))$.
