Divisibility by a prime ($pA=A$)

An abelian group $A$ is $p$-divisible when multiplication by $p$ is surjective, equivalently $A/pA=0$. This does not mean it has no $p$-torsion: $\mathbb{Q}/\mathbb{Z}$ is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.