 Subgroup generator bound for finitely generated Abelian groups

Every subgroup of an Abelian group generated by $n$ elements has at most $n$ generators. Lift it under a surjection ${\mathbb{Z}}^n\to A$. A subgroup $L\le {\mathbb{Z}}^n$ has at most $n$ generators by induction: its first-coordinate image is either zero or $d\mathbb{Z}$. In the latter case choose a lift of $d$, and subtract its multiples to reduce to the kernel in ${\mathbb{Z}}^{n-1}$. The images of these generators generate the original subgroup. The Abelian hypothesis matters: $S_{2n+2}$ is generated by a transposition and a full cycle but contains the elementary abelian group generated by $n+1$ disjoint transpositions, requiring $n+1$ generators by its vector-space dimension over ${\mathbb{F}}_2$.