Direct product of Z10 and Z2
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Definition
This group is defined in the following equivalent ways:
- It is the direct product of the cyclic group of order ten and the cyclic group of order two.
- It is the direct product of the cyclic group of order five and the Klein four-group.
- It is the direct product of the cyclic group of order five and two copies of the cyclic group of order two.
Arithmetic functions
| Function | Value | Explanation |
|---|---|---|
| order of a group | 20 | |
| exponent of a group | 10 | |
| Frattini length | 2 | |
| Fitting length | 1 | |
| derived length | 1 | |
| minimum size of generating set | 2 |
Group properties
| Property | Satisfied | Explanation |
|---|---|---|
| cyclic group | No | |
| abelian group | Yes | |
| nilpotent group | Yes |
GAP implementation
Group ID
This finite group has order 20 and has ID 5 among the groups of order 20 in GAP's SmallGroup library. For context, there are groups of order 20. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(20,5)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(20,5);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [20,5]
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 described using GAP's DirectProduct, CyclicGroup and ElementaryAbelianGroup functions, in any of these ways:
DirectProduct(CyclicGroup(10),CyclicGroup(2))
DirectProduct(CyclicGroup(5),ElementaryAbelianGroup(4))
DirectProduct(CyclicGroup(5),CyclicGroup(2),CyclicGroup(2))