# Why does Schnorr's Digital Signature scheme necessitate two prime numbers?

One of the necessary components to the Schnorr Digital Signature scheme is a pair of prime numbers p and q such that q divides p-1. However, there is never a modular inverse taken of q so why is there an extra constraint on q? Couldn't q be a number that divides p-1 but not prime?

-
However, to achieve security we need that the discrete logarithm problem in that group is hard. So the reason for the choice of $q$ (which is the group order) is that DL is believed to be hard in subgroups of prime order $q$ of $\mathbb{Z}_p^*$, where p is a safe prime, i.e. $p=2q+1$. (And also for other values than $2$, but I'm currently not sure what the conditions are.)