Subgroups of symplectic groups that contain a subsystem subgroup

Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 R*; and let ν be a symmetric equivalence relation on the index set {1,..., l, -l,..., 1} all of whose classes contain at least 3 elements. In the present paper, we prove that if a subgroup H of Γ contains the group EΓ(ν) of elementary block diagonal matrices of type ν, then H normalizes the subgroup generated by all elementary symplectic transvections Tij(ξ) ∈ H. Combined with the previous results, this completely describes the overgroups of subsystem subgroups in this case. Similar results for subgroups of GL(n, R) were established by Z. I. Borewicz and the author in the early 1980s, while for GSp(2l, R) and GO(n, R) they have been announced by the author in the late 1980s, but a complete proof for the symplectic case has not been published before. Bibliography: 74 titles. © 2008 Springer Science+Business Media, Inc.