Cyclic group:Z15

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

This group, denoted $C_{15},\mathbb {Z} _{15}$ or $\mathbb {Z} /15\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order $15$.
2. It is the direct product of the cyclic group of order three and the cyclic group of order five.
3. It is the unique group of order $15$ up to isomorphism.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 15#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	15	groups with same order
exponent of a group	15	groups with same order and exponent of a group | groups with same exponent of a group
nilpotency class	1	groups with same order and nilpotency class | groups with same nilpotency class	cyclic implies abelian
derived length	1	groups with same order and derived length | groups with same derived length	cyclic implies abelian

GAP implementation

Group ID

This finite group has order 15 and has ID 1 among the groups of order 15 in GAP's SmallGroup library. For context, there are groups of order 15. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(15,1)

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

gap> G := SmallGroup(15,1);

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

IdGroup(G) = [15,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description	Functions used
CyclicGroup(15)	CyclicGroup
DirectProduct(CyclicGroup(5),CyclicGroup(3))	CyclicGroup, DirectProduct