Direct product of cyclic group of prime-cube order and cyclic group of prime order

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

This group is defined as the external direct product of the cyclic group of prime-cube order (denoted $\mathbb{Z}_{p^3}$ or $C_{p^3}$) and the cyclic group of prime order (denoted $\mathbb{Z}_p$ or $C_p$), i.e., it is defined as $\mathbb{Z}_{p^3} \times \mathbb{Z}_p$ (or, in alternative notation, $C_{p^3} \times C_p$). It corresponds to the partition (see classification of finitely generated abelian groups and abelian group of prime power order):

$\! 4 = 3 + 1$

For a given prime $p$, it can also be defined as the unique (up to isomorphism) abelian group of order $p^4$ and exponent $p^3$.

Particular cases

Value of prime number $p$	Corresponding group
2	direct product of Z8 and Z2
3	direct product of Z27 and Z3
5	direct product of Z125 and Z5

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
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
prime-base logarithm of exponent	3	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 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	2	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	2	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	2	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	2	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	2	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

GAP implementation

Group ID

This finite group has order p^4 and has ID 5 among the group of order p^4 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^4,5)

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

gap> G := SmallGroup(p^4,5);

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

IdGroup(G) = [p^4,5]

or just do:

IdGroup(G)

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

Short descriptions

Description	Functions used	Mathematical comments
DirectProduct(CyclicGroup(p^3),CyclicGroup(p))	DirectProduct and CyclicGroup