Alice chooses a group of prime order p, with generator g.

Note that the depends on the order P of the local regression.

The group has a unique subgroup of order p, call it H.

The roots of a primitive polynomial all have order p 1.

The maximal class of a group of order p is n - 1.

A uniform presentation of the extra special groups of order p can be given as follows.

The groups F above have order p 1.

Let - the Galois field of order p the characteristic.

Since groups of order p are always regular groups, it is also a minimal such example.

There are exactly two groups of order p, namely and .