IsCyclic - Maple Help
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GroupTheory

 IsCyclic
 attempt to determine whether a group is cyclic

 Calling Sequence IsCyclic( G )

Parameters

 G - a group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(6,1\right):$
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{IsCyclic}\left(\mathrm{SmallGroup}\left(6,2\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsCyclic}\left(\mathrm{QuasicyclicGroup}\left(11\right)\right)$
 ${\mathrm{false}}$ (3)
 > $G≔\mathrm{Subgroup}\left(\left[\frac{4}{81},\frac{16}{27}\right],\mathrm{QuasicyclicGroup}\left(3\right)\right)$
 ${G}{≔}{{ℤ}}_{{{3}}^{{\mathrm{\infty }}}}$ (4)
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $G≔⟨a|{a}^{6}=1⟩$
 ${G}{≔}⟨{}{a}{}{\mid }{}{{a}}^{{6}}{}⟩$ (6)
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{true}}$ (7)
 > $G≔⟨⟨a,b⟩|⟨{a}^{3}={b}^{2},{b}^{3}⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{b}}^{{3}}{,}{{a}}^{{-3}}{}{{b}}^{{2}}{}⟩$ (8)
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsCyclic}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(4\right),\mathrm{CyclicGroup}\left(15\right)\right)\right)$
 ${\mathrm{true}}$ (10)

Compatibility

 • The GroupTheory[IsCyclic] command was introduced in Maple 2015.
 • For more information on Maple 2015 changes, see Updates in Maple 2015.

 See Also