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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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:


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:


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: