  In \cite{Pillay} and more formally in \cite{Onshuus-Pillay} it was asked
whether open subgroups of $p$-adic algebraic groups are ($p$-adic)
semialgebraic, equivalently, definable in the structure $(\mathbb Q_{p}, +,
\times)$. We give a positive answer in the commutative case. Together with
results of \cite{Prasad} this leads to a positive answer for reductive
algebraic groups.
