Let $p$ be a fixed prime in $\Bbb Z$ and $Ab$ represent the category of abelian groups. Define a functor $F : Ab \rightarrow Ab$ by $F(G) = G/pG$ and $F(f) = x + pG \rightarrow f(x) + pH$ (where $f:G \rightarrow H$ is a homomorphism)
What does $pG$ mean for an abelian group? I am familiar with the notation $aG = \{ag : g \in G\}$, but $G$ isn't specified to be an integer group which would be compatible with a prime $p$.