SUBGROUPS OF THE GENERAL LINEAR GROUP THAT CONTAIN ELEMENTARY SUBGROUP OVER A RANK 2 COMMUTATIVE RING EXTENSION

摘要

Let R = ∏ i ∈ I F i $$ prod limits_{iin I}{F}_i $$ be the direct product of fields, and let S = R d = ∏ i ∈ I F i d i $$ S=Rleft[sqrt{d}ight]=prod limits_{iin I}{F}_ileft[sqrt{d_i}ight] $$ be a rank 2 extension of R. The subgroups of the general linear group GL(2n,R), n ≥ 3, that contain the elementary group E (n, S) are described. It is shown that for every such a subgroup H there exists a unique ideal A ⊴ R such that E (n, S)E(2n,R,A) ≤ H ≤ N~(GL(2n,R))(E (n, S)E(2n,R,A)).