# Cyclic group:Z5

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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
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## Definition

This group, denoted $C_5, \Z_5, \Z/5\Z$, is defined in the following equivalent ways:

## Arithmetic functions

Function Value Explanation
order 5
exponent 5
derived length 1 The group is abelian.
Frattini length 1
Fitting length 1

## Group properties

Property Satisfied Explanation
cyclic group Yes
abelian group Yes
nilpotent group Yes
solvable group Yes

## GAP implementation

### Group ID

This finite group has order 5 and has ID 1 among the groups of order 5 in GAP's SmallGroup library. For context, there are groups of order 5. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(5,1)

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

gap> G := SmallGroup(5,1);

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

IdGroup(G) = [5,1]

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(5)