Nu of Prime Number is 1

Theorem
Let $p$ be a prime number.

Then:
 * $\nu \paren p = 1$

where $\nu$ denotes the $\nu$ function: the number of types of group of a given order.

Proof
Let $G_1$ and $G_1$ be groups of order $p$.

From Prime Group is Cyclic, $G_1$ and $G_1$ are both cyclic groups.

From Cyclic Groups of Same Order are Isomorphic, $G_1$ and $G_1$ are isomorphic.

Thus by definition, $G_1$ and $G_1$ are of the same type.

Hence the result.