Cyclic group of prime-square order

Definition

Let $p$ be a prime number. The cyclic group of order $p^{2}$, denoted $C_{p^{2}},\mathbb {Z} _{p^{2}},\mathbb {Z} /p^{2}\mathbb {Z}$, is defined as the cyclic group with $p^{2}$ elements.

This is the cyclic group corresponding to the partition $2$ of $2$.

Particular cases

prime number $p$	corresponding cyclic group of prime-square order
2	cyclic group:Z4
3	cyclic group:Z9
5	cyclic group:Z25
7	cyclic group:Z49

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	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	1	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	1	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	1	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	1	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	1	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 1 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^2,1)

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

gap> G := SmallGroup(p^2,1);

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^2,1]

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
CyclicGroup(p^2)	CyclicGroup