Cyclic group:Z7

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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_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)