Cyclic group:Z20

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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)