CyclicGroup - Maple Help

GroupTheory

 CyclicGroup

 Calling Sequence CyclicGroup( n ) CyclicGroup( n, s )

Parameters

 n - algebraic; understood to be a positive integer or infinity s - (optional) equationof the form form="fpgroup" or form="permgroup" (the default)

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{CyclicGroup}\left(14\right)$
 ${{C}}_{{14}}$ (1)
 > $\mathrm{CyclicGroup}\left(14,'\mathrm{form}'="permgroup"\right)$
 ${{C}}_{{14}}$ (2)
 > $\mathrm{CyclicGroup}\left(14,'\mathrm{form}'="fpgroup"\right)$
 $⟨{}{g}{}{\mid }{}{{g}}^{{14}}{}⟩$ (3)
 > $\mathrm{CyclicGroup}\left(\mathrm{\infty }\right)$
 $⟨{}{\mathrm{g0}}{}{\mid }{}{}⟩$ (4)
 > $\mathrm{Degree}\left(\mathrm{CyclicGroup}\left(12\right)\right)$
 ${12}$ (5)
 > $\mathrm{Degree}\left(\mathrm{CyclicGroup}\left(12,':-\mathrm{mindegree}'\right)\right)$
 ${7}$ (6)
 > $\mathrm{CyclicGroup}\left({2}^{7}{3}^{7}{5}^{7}\right)$
 > $G≔\mathrm{CyclicGroup}\left({2}^{7}{3}^{7}{5}^{7},':-\mathrm{mindegree}'\right)$
 ${G}{≔}{{C}}_{{21870000000}}$ (7)
 > $\mathrm{Degree}\left(G\right)$
 ${80440}$ (8)
 > $G≔\mathrm{CyclicGroup}\left(2k+4\right)$
 ${G}{≔}{{\mathbf{C}}}_{{2}{}{k}{+}{4}}$ (9)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsSimple}\left(G\right)$
 $\left\{\begin{array}{cc}{\mathrm{true}}& \left({2}{}{k}{+}{4}\right){::}{\mathrm{prime}}\\ {\mathrm{false}}& {\mathrm{otherwise}}\end{array}\right\$ (11)

Compatibility

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