A cyclic group is an abelian group generated by a single element. The CyclicGroup command returns a group, either as a permutation group, or a group defined by a generator and a relator, isomorphic to a cyclic group of order n.

By default, a permutation group is returned if n is finite, but you can specify that the cyclic group of order n be constructed as a finitely presented group by passing the option form = "fpgroup".

If n = infinity, then a finitely presented group is returned. It is an error to specify form = permgroup if the argument n is equal to infinity.

You can use the mindegree option to create cyclic permutation groups of much larger order than would be possible without this option. By default, mindegree = false but, if you pass mindegree = true (or just mindegree), then a permutation group of minimal degree which is cyclic of the indicated order is returned.

If n is neither infinity nor a positive integral constant, then a symbolic group representing a cyclic group of order equal to the expression n (which is taken to represent a positive integer) is returned.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$\mathrm{CyclicGroup}\left(14\right)$

$C_{14}$

$\mathrm{CyclicGroup}\left(14\,'\mathrm{form}'=''permgroup''\right)$

$C_{14}$

$\mathrm{CyclicGroup}\left(14\,'\mathrm{form}'=''fpgroup''\right)$

$⟨g\mid g^{14}⟩$

$\mathrm{CyclicGroup}\left(\mathrm{\infty }\right)$

$⟨\mathrm{g0}\mid ⟩$

$\mathrm{Degree}\left(\mathrm{CyclicGroup}\left(12\right)\right)$

$12$

$\mathrm{Degree}\left(\mathrm{CyclicGroup}\left(12\,':-\mathrm{mindegree}'\right)\right)$

$7$

$\mathrm{CyclicGroup}\left(2^7{3}^7{5}^7\right)$

Error, (in GroupTheory:-CyclicGroup) object too large in seq

$G≔\mathrm{CyclicGroup}\left(2^7{3}^7{5}^7\,':-\mathrm{mindegree}'\right)$

$G≔C_{21870000000}$

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

$80440$

$G≔\mathrm{CyclicGroup}\left(2k+4\right)$

$G≔{\mathbf{C}}_{2k+4}$

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

$\mathrm{true}$

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

$\left\{\begin{array}{cc}\mathrm{true}& \left(2k+4\right)::\mathrm{prime}\\ \mathrm{false}& \mathrm{otherwise}\end{array}\right.$

The GroupTheory[CyclicGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.