Cyclic group:Z20
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Contents
Definition
This group is defined in the following equivalent ways:
- It is the cyclic group of order
.
- 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 :
gap> G := SmallGroup(20,2);
Conversely, to check whether a given group 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)