Let \(G\) be a group whose order is equal to \(p\), a prime number. Then \(G\) is isomorphic to the cyclic group \(C_p\) of order \(p\).

Proof

Pick any non-identity element \(g\) of the group.

By Lagrange’s theorem, the subgroup generated by \(g\) has size \(1\) or \(p\) (since \(p\) was prime). But it can’t be \(1\) because the only subgroup of size \(1\) is the trivial subgroup.

Hence the subgroup must be the entire group.