ON THE OVERGROUPS OF SL(1, K) IN GL(r, F )

Abstract

For some pair of division rings K and F with K ⊃ F and dim_F K= r , we want to determine the overgroups of K^∗ = SL(1, K) in GL(r, F) and obtain the maximal subgroups of GL(r, F).  Let R stand for real number field, C for complex one and Q for the skew-field of quaternions.  All the overgroups of C^∗ = SL(1, C) in GL(2, R) and Q^∗ = SL(1, Q) in GL(2, C) are found in this paper.