 Primary torsion subgroup at a prime

For an abelian group $A$, its $p$-primary torsion subgroup is $A\{p\}=\{a:p^{r}a=0\text{ for some }r\ge 0\}$. It is zero exactly when $A[p]=0$, since a nonzero element of $p$-power order has a multiple of order $p$. Multiplication by an integer prime to $p$ is an automorphism on it.