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

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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