= Subgroup generator bound for finitely generated Abelian groups
{title2=$d(B)\leq n\quad(B\leq A,\ d(A)\leq n)$}
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> $L\leq\mathbb Z^n$ 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 $\mathbb F_2$.
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