Cyclic group:Z16
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Contents
Definition
The cyclic group of order sixteen, denoted or sometimes
, is the cyclic group having
elements. In other words, it is the quotient of the group of integers
by the subgroup
of multiples of
.
It is given by the presentation:
where is the identity element.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Group properties
Property | Satisfied? | Explanation |
---|---|---|
cyclic group | Yes | |
homocyclic group | Yes | |
metacyclic group | Yes | |
abelian group | Yes |
GAP implementation
Group ID
This finite group has order 16 and has ID 1 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(16,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(16,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [16,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Alternative descriptions
Description | Functions used |
---|---|
CyclicGroup(16) | CyclicGroup |