Is every discrete, torsion and divisible group $\sigma-$compact?

Question

Let $G$ be a discrete, torsion and divisible group. Is $G$, $\sigma-$compact?

Answer 1

Take $G$ be a torsion divisible abelian group. Let $H$ be an uncountable direct sum of $G$ with itself. Endow $H$ with discrete topology. The result is divisible torsion group which is not sigma-compact.