How can prime factor of $q-1$ divide prime $q$? In this paper - &quot;A Subexponential Algorithm for the Discrete Logarithm Problem&quot;, author mentions (page 56),

For each $p_l^{e_l} | q$ proceed....

$p_l^{e_l}$  is one of the prime factors of $q-1$, and $q$ is a prime.
How can any number divide prime? Am I missing something here?
Edit1: $q$ is prime, so we can't get any non-trivial factor of $q$. Keeping this in mind, can anyone explain how to proceed with the steps mentioned in the paper. One thing to keep in mind is that $q$ has to be prime to be used in Diffie-Hellman system.
For arbitrary positive integer $n$ we have:


  $$\gcd(n,n+1)=1$$


So $q-1$ and $q$ haven't any nontrivial common factor.