# Cyclic group:Z7

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## Definition

This group, denoted $C_7, \Z_7, \Z/7\Z$ is defined as the cyclic group of order $7$, i.e., the group of integers modulo n where $n = 7$. Equivalently, it is the additive group of the field of seven elements.

## Arithmetic functions

Function Value Explanation
order 7
exponent 7
Frattini length 1
Fitting length 1
subgroup rank 1
rank as p-group 1

## Group properties

Property Satisfied Explanation
cyclic group Yes
abelian group Yes
homocyclic group Yes
elementary abelian group Yes

## GAP implementation

### Group ID

This finite group has order 7 and has ID 1 among the groups of order 7 in GAP's SmallGroup library. For context, there are groups of order 7. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(7,1)

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

gap> G := SmallGroup(7,1);

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

IdGroup(G) = [7,1]

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(7)