Braid group action via GL n (q) and U n (q), and Galois realizations

We determine the braid group action on generating systems of a group that is the semi-direct product of a finite vector space with a group of scalars. This leads to Galois realizations of certain groups GL _n (q) and PU _n (q). Dedicated to Prof. J. G. Thompson Partially supported by an NSA grant. This work was done while the author was a fellow of the Institute for Advanced Studies in Jerusalem.