Editing P-group (section)

===Up to ''p''<sup>3</sup>===
The trivial group is the only group of order one, and the cyclic group C<sub>''p''</sub> is the only group of order ''p''. There are exactly two groups of order ''p''<sup>2</sup>, both abelian, namely C<sub>''p''<sup>2</sup></sub> and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. For example, the cyclic group C<sub>4</sub> and the [[Klein four-group]] ''V''<sub>4</sub> which is C<sub>2</sub>&nbsp;×&nbsp;C<sub>2</sub> are both 2-groups of order 4.

There are three abelian groups of order ''p''<sup>3</sup>, namely C<sub>''p''<sup>3</sup></sub>, C<sub>''p''<sup>2</sup></sub>&nbsp;×&nbsp;C<sub>''p''</sub>, and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. There are also two non-abelian groups.

For ''p''&nbsp;≠&nbsp;2, one is a semi-direct product of C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub> with C<sub>''p''</sub>, and the other is a semi-direct product of C<sub>''p''<sup>2</sup></sub> with C<sub>''p''</sub>. The first one can be described in other terms as group UT(3,''p'') of unitriangular matrices over finite field with ''p'' elements, also called the [[Heisenberg group#Heisenberg group modulo an odd prime p|Heisenberg group mod ''p'']].

For ''p''&nbsp;=&nbsp;2, both the semi-direct products mentioned above are isomorphic to the [[dihedral group]] Dih<sub>4</sub> of order 8. The other non-abelian group of order 8 is the [[quaternion group]] Q<sub>8</sub>.