UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

Given a finite dimensional F-central simple algebra A = M-n(D), the connection between the Frattini subgroup Phi(A*) and Phi(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* boolean AND Phi(A*), it is shown that Phi(F*) Z(A') subset of G subset of (boolean AND(p) F*(p)) Z(A') where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Phi(A*) is also calculated for some particular division rings D.