Cyclic group:Z16

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Definition

The cyclic group of order sixteen is the cyclic group having $16$ elements. In other words, it is the quotient of the group of integers $\mathbb{Z}$ by the subgroup $16\mathbb{Z}$ of multiples of $16$.

It is given by the presentation:

$⟨x\mid x^{16}=e⟩$

where $e$ 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

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	16	groups with same order and exponent of a group | groups with same prime-base logarithm of order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	4	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length	4	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length	Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set	1	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	1	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group	same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group	1	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group	1	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class	1		The group is a nontrivial abelian group
derived length	1		The group is a nontrivial abelian group
Fitting length	1		The group is a nontrivial abelian group -

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 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 $G$:

gap> G := SmallGroup(16,1);

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

IdGroup(G) = [16,1]

or just do:

IdGroup(G)

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