= Finite generation detected by a prime quotient
{title2=$A\text{ finite}\iff A/pA=0\text{ for some prime }p$}
The <Fundamental theorem of finitely generated abelian groups> writes $A=\mathbb Z^r\oplus F$ with $F$ finite. Reduction modulo a prime contains $(\mathbb Z/p\mathbb Z)^r$, so a vanishing quotient forces $r=0$. Conversely multiplication by a prime coprime to $|F|$ is bijective on $F$. Finite generation matters: the nonzero <divisible group> $(\mathbb Q,+)$ satisfies $p\mathbb Q=\mathbb Q$ for every prime and is not finitely generated, since any finite list of rational generators has a common denominator.
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