Groups of prime-square order

Let $p$ be a prime number. There are two groups of order $p^{2}$, namely:

Group name	Symbols typically used for group	Description of group	Probability in cohomology tree probability distribution
cyclic group of prime-square order	$\mathbb {Z} _{p^{2}},C_{p^{2}},\mathbb {Z} /p^{2}\mathbb {Z}$	cyclic group whose order is $p^{2}$	$1-1/p$
elementary abelian group of prime-square order	$E_{p^{2}}$, $\mathbb {Z} _{p}\times \mathbb {Z} _{p}$, $C_{p}\times C_{p}$	elementary abelian group whose order is $p^{2}$	$1/p$

Both of these are abelian groups.

For a proof that these are the only groups of prime-square order, see classification of groups of prime-square order.

Particular cases

Value of prime number $p$	Cyclic group of order $p^{2}$	Elementary abelian group of order $p^{2}$
2	cyclic group:Z4	Klein four-group
3	cyclic group:Z9	elementary abelian group:E9
5	cyclic group:Z25	elementary abelian group:E25