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

Let $p$ be a prime number. This group is defined in the following equivalent ways:

1. It is the external direct product of two copies of the cyclic group of prime-square order, i.e., it is the group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2}$.
2. It is the homocyclic group of order $p^4$ and exponent $p^2$.

Particular cases

Value of prime number $p$	Corresponding group
2	direct product of Z4 and Z4
3	direct product of Z9 and Z9
5	direct product of Z25 and Z25

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	2	groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length	2	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

"p^" can not be assigned to a declared number type with value 4.

and has ID 2 among the groups of order p^4 in GAP's SmallGroup library. For context, there are groups of order p^4. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^4,2)

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

gap> G := SmallGroup(p^4,2);

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^4,2]

or just do:

IdGroup(G)

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

"p^" can not be assigned to a declared number type with value 4.

Short descriptions

Description	Functions used	Mathematical comments
DirectProduct(CyclicGroup(p^2),CyclicGroup(p^2))	DirectProduct and CyclicGroup