# Direct product of Z10 and Z2

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## Definition

This group is defined in the following equivalent ways:

1. It is the direct product of the cyclic group of order ten and the cyclic group of order two.
2. It is the direct product of the cyclic group of order five and the Klein four-group.
3. 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 Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 20 groups with same order
exponent of a group 10 groups with same order and exponent of a group | groups with same exponent of a group
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length
Fitting length 1 groups with same order and Fitting length | groups with same Fitting length
derived length 1 groups with same order and derived length | groups with same derived length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set

## 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 5 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 $G$:

gap> G := SmallGroup(20,5);

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

IdGroup(G) = [20,5]

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))