Cyclic group:Z8

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The cyclic group of order eight, denoted $C_{8}$, $\mathbb {Z} _{8}$, or $\mathbb {Z} /8\mathbb {Z}$, is defined as the cyclic group of order eight, i.e., it is the quotient of the group of integers $\mathbb {Z}$ by the subgroup $8\mathbb {Z}$ of multiples of eight.

Multiplication table

The following image shows visually the multiplication table of this group. The $i$th column/row represents the element $g^{i-1}$ (i.e. the first row/column represent the identity), in the presentation $\langle g\mid g^{8}=e\rangle$ of this group.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#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)	8	groups with same order
prime-base logarithm of order	3	groups with same prime-base logarithm of order
max-length of a group	3		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	3		chief length equals prime-base logarithm of order for group of prime power order
composition length	3		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	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	3	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	3	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
abelian group	Yes
homocyclic group	Yes
metacyclic group	Yes
group of prime power order	Yes
nilpotent group	Yes

Automorphism group

Further information: automorphism group of cyclic group

The automorphism group of $\mathbb {Z} _{8}$ is the Klein four-group. In fact, this is the smallest cyclic group whose automorphism group is not cyclic.

GAP implementation

Group ID

This finite group has order 8 and has ID 1 among the groups of order 8 in GAP's SmallGroup library. For context, there are groups of order 8. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(8,1)

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

gap> G := SmallGroup(8,1);

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

IdGroup(G) = [8,1]

or just do:

IdGroup(G)

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

Other descriptions

The group can be described using GAP's CyclicGroup function:

CyclicGroup(8)