Cyclic group of prime-square order

Definition

Let $p$ be a prime number. The cyclic group of order $p^{2}$ , denoted $C_{p^{2}}, Z_{p^{2}}, Z / p^{2} Z$ , 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

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