This is a topic in classical algebraic K-Theory. I will recall definitions of elementary linear group, elementary symplectic group, linear transvection group, and symplectic transvection group. These group= s have natural action on the set of unimodular elements. I will briefly discuss how bijections between orbit spaces of unimodular elements under different group actions are established. Finally, I will talk about an application of these results, namely improving injective stability bound for the K1 group.

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Last updated: 09 Oct 2024