Is modulus a prime number important for non-symmetric cryptology? From this link
Generation of a cyclic group of prime order
we know how to generate a prime order group.

 

This illustrates why a prime order group is important.

But why is modulus a prime number also important?

In the theory of RSA, the modulus $N=pq$, and $\operatorname{totient}(N)=(p-1)(q-1)$， if we can find a factor $p'$ of $p-1$, then we can claim there is a $p'$ order sub-group? But it is not a modulus prime group, the $N$ is a composite number, is such a group useful?
Yes, such a group is useful. In particular, when $N= p\cdot q$ where $p,q$ are both strong primes (i.e. $p=2p'+1,q=2q'+1$ where $p',q'$ are also prime numbers). 

Discrete logarithm in the prime order cyclic groups (order-$p'$ and order-$q'$) is believed to be hard. Such a group is called a group with hidden order.  

A few examples of application: Fujisaki and Okamoto used this group to construct a commitment scheme. It has also been used in constructing a zero-knowledge proof of an integer falls within an interval (range proof). 

Note that security does not hold for any arbitrary $N$ (even if it has only two large prime factors), see this paper.