# GAP:CyclicGroup

From Groupprops

This article is about a GAP function.

## Definition

### Function type

The function takes as input a positive integer and outputs a group. An optional filter can be provided that controls the form in which the group is constructed and stored.

### Behavior

The function is invoked by:

`CyclicGroup([<filt>, ]n)`

where is a positive integer. Here are the possibilities:

Nature of input | Output |
---|---|

No filter, a positive integer | cyclic group (specifically, finite cyclic group) of order constructed as a PcGroup, i.e., in terms of a polycyclic series |

IsPermGroup as filter, a positive integer | cyclic group of order constructed as a permutation group on a set of size (via the Cayley embedding, so its generator is a -cycle) |

IsMatrixGroup as filter, a positive integer | cyclic group of order constructed as permutation matrices (again via Cayley embedding) |

## Related functions

## Examples of usage

gap> G := CyclicGroup(5); <pc group of size 5 with 1 generators> gap> H := CyclicGroup(8); <pc group of size 8 with 3 generators> gap> Subgroups(H); [ Group([ <identity> of ... ]), Group([ f3 ]), Group([ f2 ]), Group([ f1, f2, f3 ]) ] gap> K := CyclicGroup(IsPermGroup,9); Group([ (1,2,3,4,5,6,7,8,9) ]) gap> L := CyclicGroup(IsMatrixGroup,3); Group([ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]) gap> Set(L); [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] gap> CyclicGroup(0); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `CyclicGroupCons' on 2 arguments called from CyclicGroupCons( IsPcGroup, arg[1] ) called from <function>( <arguments> ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk>