A group $G$ is said to be finitely generated if it can be generated by a finite subset of its elements.

The IsFinitelyGenerated( G ) command returns true if the group G is finitely generated, and returns the value false otherwise.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{IsFinitelyGenerated}\left(\mathrm{PSp}\left(4\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsFinitelyGenerated}\left(\mathrm{Symm}\left(30\right)\right)$

$\mathrm{true}$

$\mathrm{IsFinitelyGenerated}\left(\mathrm{QuasicyclicGroup}\left(37\right)\right)$

$\mathrm{false}$

$\mathrm{IsFinitelyGenerated}\left(⟨⟨a\,b⟩|a^{-2}·b^{-2}·a^3·b^3=1⟩\right)$

$\mathrm{true}$