Cyclic group:Z20

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Definition

This group is defined in the following equivalent ways:

1. It is the cyclic group of order $20$.
2. It is the direct product of the cyclic group of order five and cyclic group of order four.

Arithmetic functions

Function Value Explanation
order 20
exponent 20
Frattini length 2
Fitting length 1
derived length 1

Group properties

Property Satisfied Explanation
cyclic group Yes
abelian group Yes
metacyclic group Yes

GAP implementation

Group ID

This finite group has order 20 and has ID 2 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(20,2)

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

gap> G := SmallGroup(20,2);

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

IdGroup(G) = [20,2]

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(20)