Prime Groups of Same Order are Isomorphic

Theorem
Two prime groups of the same order are isomorphic to each other.

Proof
Let $G_1$ and $G_2$ be prime groups, both of finite order $p$.

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

The result follows directly from Cyclic Groups of Same Order are Isomorphic.