Cyclic group:Z21

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Definition

This group, denoted $C_{21},\mathbb {Z} _{21},\mathbb {Z} /21\mathbb {Z}$ is defined as the cyclic group of order $21$, i.e., the group of integers modulo n where $n=21$.

This is the only abelian group of order 21. It is not the only group of this order, other group of order 21 is non-abelian.

Arithmetic functions

Function	Value	Explanation
order	21

Group properties

Property	Satisfied	Explanation
cyclic group	Yes
abelian group	Yes
homocyclic group	No

GAP implementation

Group ID

This finite group has order 21 and has ID 2 among the groups of order 21 in GAP's SmallGroup library. For context, there are groups of order 21. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(21,2)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(21,2);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [21,2]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The group can be constructed using GAP's CyclicGroup function:

CyclicGroup(21)