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For a finite group $G$ with a symmetric generating set $S$, its diameter isEquivalently, it is the diameter of the Cayley graph ${\mathrm{Cay}}_S(G)$.

A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.

The hidden subgroup problem for a group $G$ is specified by an oracle $f:G\to S$ that hides an unknown subgroup $H\le G$: it is constant on each left coset of $H$ and takes distinct values on distinct cosets. The task is to determine $H$, usually by finding a generating set for it.