Template:Cyclic group of prime order

Definition

This group, denoted {{{3}}} is defined as the cyclic group of order {{{2}}}. Equivalently, it is the additive group of the [[field:F{{{1}}}|field of {{{4}}} elements]].

Note that {{{1}}} is prime, and thus by the classification of groups of prime order, this is the only [[groups of order {{{1}}}|group of order {{{1}}}]].

Arithmetic functions

Function	Value	Explanation
order	{{{1}}}
exponent	{{{1}}}
Frattini length	1
Fitting length	1
subgroup rank	1
rank as p-group	1	{{{1}}} is a prime number

Group properties

Property	Satisfied	Explanation
cyclic group	Yes
abelian group	Yes	Cyclic implies abelian
nilpotent group	Yes	It is abelian by above, and abelian implies nilpotent
homocyclic group	Yes	Cyclic groups are homocyclic
elementary abelian group	Yes
simple group	Yes	Cyclic groups of prime order are simple

GAP implementation

Group ID

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

SmallGroup({{{1}}},1)

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

gap> G := SmallGroup({{{1}}},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) = [{{{1}}},1]

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({{{1}}})