Abstract:
The aim of this thesis is to show a bijection between the orbit spaces of unimodular rows under the action of the elementary linear group and the orbit spaces of unimodular rows under the action of the elementary symplectic group. Also established a relative version of it with respect to an ideal. Then generalized this result and shown that the orbit space of unimodular rows of a projective module under the action of the group of elementary transvections, is in bijection with the orbit space of unimodular rows of a projective module under the action of the group of elementary symplectic transvections with respect to an alrternating form. Specific equalities are used to improve the injective stability bound for K1Sp(R) and Sp(Q, (,)) / E TransSp(Q, (,)).
