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A group is finitely generated when it has a finite generating set of a group.

Let $G$ have a generating set of a group with $r$ elements. Every subgroup $H$ of index $n$ gives a transitive group action of $G$ on the $n$ left cosets of $H$, hence a group homomorphism ${\rho }_H:G\to S_n$ after the cosets are labelled. Conversely, $H$ is the stabilizer subgroup of one point in this action.

A homomorphism $G\to S_n$ is determined by the images of the $r$ generators, so there are at most $(n!{)}^r$ such homomorphisms and at most $n(n!{)}^r$ point stabilizers. Therefore the finiteness of subgroups of a fixed finite index gives only finitely many subgroups of index $n$.

The rank $d(G)$ of a finitely generated group is the minimum cardinality of a generating set of a group of $G$.