FRATTINI SUBGROUP OF THE UNIT GROUP OF CENTRAL SIMPLE ALGEBRAS.

Given an F-central simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ⋂p∤deg(A) F*p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k. [ABSTRACT FROM AUTHOR]