Cyclic group:Z81

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Definition

This group is defined as the cyclic group of order $81$. It is denoted as $C_{81}$, ${\mathbb{Z}}_{81}$ or $\mathbb{Z}/81\mathbb{Z}$.

Arithmetic functions

Function	Value	Explanation
order	81
exponent	81
Frattini length	4
derived length	1
Fitting length	1
minimum size of generating set	1

Group properties

Properties	Satisfied	Explanation
cyclic group	Yes
abelian group	Yes
metacyclic group	Yes
homocyclic group	Yes
group of prime power order	Yes
nilpotent group	Yes

GAP implementation

Group ID

This finite group has order 81 and has ID 1 among the groups of order 81 in GAP's SmallGroup library. For context, there are groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,1)

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

gap> G := SmallGroup(81,1);

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

IdGroup(G) = [81,1]

or just do:

IdGroup(G)

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

Other descriptions

This group can be defined using GAP's CyclicGroup function:

CyclicGroup(81)