Compact open subgroup of a locally compact abelian group

Question

Let $L$ be a compact open subgroup of locally compact abelian group $G$. Is $nL=\{nx;x\in L\}$ an open subgroup of $G$?

Answer 1

It seems that the answer is negative. For instance, $G=L=\mathbb Z_2^\omega$ is a compact group, but the set $2L=\{e\}$ is not open in $G$.