Abstract
Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S⩾2, R is generated by its invertible elements, and the ideal of R generated by the differences ε−1 for all invertible ɛ coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net б of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e(б)⩽P⩽G(б), where G is the net subgroup of (б) and E(б) is the subgroup generated by the transvections in G(б). It is shown that E(б) is a normal subgroup of G(б). The factor group G(б/E(б)) is studied. The case of the special linear group is also considered.
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