Why is it that in DSA, the order of the subgroup, $q$, is chosen such that it divides $p - 1$? Consider the DSA key generation:


A large prime $p$ is chosen;
A smaller modulus $q$ is chosen such that $p - 1$ is a multiple of $q$;
A generator $g$ s.t. $\operatorname{ord}_p(g) = q$ is chosen.


My question is -- why do we require that $p - 1$ be a multiple of $q$?  Is there any underlying mathematics I'm missing?  Thanks.
As was noted in the comments the reason for $q$ having to divide $p-1$ is Lagrange's theorem:


  Lagrange's theorem [...], states that
  for any finite group $G$, the order (number of elements) of every
  subgroup $H$ of $G$ divides the order of $G$.


In the case of DSA we are working in subgroups of $\mathbb F_p$ (which has order $p-1$). By Lagrange's theorem every subgroup has to have an order $q$ such that $q$ divides $p-1$.