# Direct product of cyclic group of prime-square order and elementary abelian group of prime-square 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

Let $p$ be a prime number.

This group is defined in the following equivalent ways:

1. $\mathbb{Z}_{p^2} \times E_{p^2}$: The external direct product of the cyclic group of prime-square order and the elementary abelian group of prime-square order.
2. $\mathbb{Z}_{p^2} \times \mathbb{Z}_p \times \mathbb{Z}_p$: The external direct product of the cyclic group of prime-square order and two copies of the cyclic group of prime order.

This group is thus the abelian group of prime power order corresponding to the partition (see also structure theorem for finitely generated abelian groups): $\! 2 + 1 + 1$

## Particular cases

Value of prime number $p$ Corresponding group
2 direct product of Z4 and V4
3 direct product of Z9 and E9

## 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 3 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 3 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 3 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 3 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 3 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 11 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^4,11)

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

gap> G := SmallGroup(p^4,11);

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^4,11]

or just do:

IdGroup(G)

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

The exception is the case $p = 2$, in which case the group is $(p^4,10)$.