Nu of Prime Number is 1

Theorem

Let $p$ be a prime number.

Then:

$\map \nu p = 1$

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

Proof

Let $G_1$ and $G_2$ be groups of order $p$.

From Prime Group is Cyclic, $G_1$ and $G_2$ are both cyclic groups.

From Cyclic Groups of Same Order are Isomorphic, $G_1$ and $G_2$ are isomorphic.

Thus by definition, $G_1$ and $G_2$ are of the same type.

Hence the result.

$\blacksquare$

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.3$