Template:Cyclic group of twice prime order

Definition

This group, denoted {{{3}}}, is defined in the following equivalent ways:

1. It is a cyclic group of order {{{1}}}.
2. It is the direct product of the cyclic group:Z2 and the [[cyclic group:Z{{{2}}}]].

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out [[groups of order {{{1}}}#Arithmetic functions]]

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	{{{1}}}	groups with same order"{{{" can not be assigned to a declared number type with value 1.
exponent of a group	{{{1}}}	groups with same order and exponent of a group"{{{" can not be assigned to a declared number type with value 1. | groups with same exponent of a group"{{{" can not be assigned to a declared number type with value 1.
nilpotency class	1	groups with same order and nilpotency class"{{{" can not be assigned to a declared number type with value 1. | groups with same nilpotency class	cyclic implies abelian
derived length	1	groups with same order and derived length"{{{" can not be assigned to a declared number type with value 1. | groups with same derived length	cyclic implies abelian

GAP implementation

Group ID

This finite group has [[groups of order {{{1}}}|order {{{1}}}]] and has ID 2 among the groups of order {{{1}}} in GAP's SmallGroup library. For context, there are groups of order {{{1}}}. It can thus be defined using GAP's SmallGroup function as:

SmallGroup({{{1}}},2)

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

gap> G := SmallGroup({{{1}}},2);

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

IdGroup(G) = [{{{1}}},2]

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({{{1}}})	CyclicGroup
DirectProduct(CyclicGroup({{{2}}}),CyclicGroup(2))	CyclicGroup, DirectProduct