Groups of prime-square order

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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
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
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

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

Arithmetic functions