On groups generated by three-dimensional special unitary groups II

Abstract

We shall determine in this paper groups of types D n , E 6 , E 7 and E 8 generated by SU (3, q)'s, q odd, q > 3. These groups are defined in Phan (1975). [We shall refer to this paper as I]. Acquaintance with the results of I is assumed. The identification of groups of type D 4 is similar to that of SU (n, q). We actually construct an isomorphism from the universal group of type D 4 onto Spin + (8, q). This direct approach does not appear to be feasible for groups of type D n with n ≧ 5. Fortunately Wong's recent result (1974) is applicable here. But his theorem requires that the characteristic of the field be odd; hence unlike the unitary case, we assume that q is odd and q 3. Using Wong's theorem, we proceed to show by induction that groups of type D n are homomorphic images of Spin + (2n, q) or Spin − (2n, q) according as n is even or n is odd.