the vanishing of sk can be interpreted as saying that k is generated by the image of gl in gl. when this fails, one can ask whether k is generated by the image of gl. for a dedekind domain, this is the case: indeed, k is generated by the images of gl and sl in gl. the subgroup of sk generated by sl may be studied by mennicke symbols. for dedekind domains with all quotients by maximal ideals finite, sk is a torsion group.