Elementary abelian group of prime-square order

Definition

Let $p$ be a prime number. We define the group $E_{p^2}$ as the elementary abelian group whose order is $p^2$. We define it as the external direct product of two copies of the group of prime order $\mathbb{Z}/p\mathbb{Z}$.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
prime-base logarithm of order	2	groups with same prime-base logarithm of order
max-length of a group	2		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	2		chief length equals prime-base logarithm of order for group of prime power order
composition length	2		composition length equals prime-base logarithm of order for group of prime power order
prime-base logarithm of exponent	1	groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length	1	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^2 and has ID 2 among the group of order p^2 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^2,2)

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

gap> G := SmallGroup(p^2,2);

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^2,2]

or just do:

IdGroup(G)

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