## Finding cyclic group $\mathbb{Z}^*_n$ of prime order

It can be proven that multiplicative group of integers modulo N defined as

$$\mathbb{Z}^*_N = \{ i∈Z : 1≤i≤N−1\; AND\; gcd(i,N)=1 \}$$

is cyclic for a prime N and that if it is of prime order, then every non-identity element in the group is a generator of this group.

How can I find such N?

I wrote a simple program and brute-forced for i $$\mathbb ∈\; [1, 300 000]$$ and found no group of prime order so far.

## $\mathbb{Z}_m \times \mathbb{Z}_n \cong \mathbb{Z}_{m’} \times \mathbb{Z}_{n’}?$

Suppose $$\mathbb{Z}_m \times \mathbb{Z}_n$$ is isomorphic to $$\mathbb{Z}_{m’} \times \mathbb{Z}_{n’}$$ as groups, where $$m$$ divides $$n$$ and $$m’$$ divides $$n’$$. Does that mean $$m=m’$$ and $$n=n’$$?