Ramification in the Inverse Galois Problem

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as group an extension rational numbers in which only predetermined set primes may ramify. After presenting new results regarding extensions single prime ramifies, we move to studying more complex situation multiple from arbitrary size then continue by examining conjecture Harbater that minimal number generators tame, is bounded above sum constant and logarithm product ramified primes. prove validity Harbater's cases, including where restrict our attention containing nilpotent subgroup index 1 , 2 or 3. also derive some consequences are implied truth this conjecture.