A group $G$ is cyclic if it can be generated by a single element.

The IsCyclic( G ) command attempts to determine whether the group G is cyclic.  It returns true if G is cyclic and returns false otherwise. The command may return FAIL on (most) finitely presented groups.

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

$G≔\mathrm{SmallGroup}\left(6\,1\right)\:$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{false}$

$\mathrm{IsCyclic}\left(\mathrm{SmallGroup}\left(6\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(\mathrm{QuasicyclicGroup}\left(11\right)\right)$

$\mathrm{false}$

$G≔\mathrm{Subgroup}\left(\left[\frac{4}{81}\,\frac{16}{27}\right]\,\mathrm{QuasicyclicGroup}\left(3\right)\right)$

$G≔{\mathbb{Z}}_{3^{\mathrm{\infty }}}$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{true}$

$G≔⟨a|a^6=1⟩$

$G≔⟨a\mid a^6⟩$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{true}$

$G≔⟨⟨a\,b⟩|⟨a^3=b^2\,b^3⟩⟩$

$G≔⟨a\,b\mid b^3\,a^{-3}{b}^2⟩$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(4\right)\,\mathrm{CyclicGroup}\left(15\right)\right)\right)$

$\mathrm{true}$

The GroupTheory[IsCyclic] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.