= Bounded-exponent finitely generated nilpotent group order bound
Let $G$ be a <nilpotent group> of class at most $s$ in which every element has order at most $r$. A subgroup generated by $k$ elements has order at most
$$
r^{sk^s}.
$$
Indeed, collection by the <lower central series> expresses every element as an ordered product of simple <group commutators> in the generators of weights at most $s$. There are at most $k+k^2+\cdots+k^s\leq sk^s$ such commutators, and each exponent may be reduced to one of at most $r$ values.
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