LET D BE A DIVISION ALGEBRA FINITE-DIMENSIONAL OVER ITS CENTER F. PUT D∗=D \ { 0} AND DENOTE BY D′ THE COMMUTATOR SUBGROUP OF D∗. LET G(D) =D∗ / NRDD (D∗) D′ WHERE NRDD (D∗) IS THE IMAGE OF D∗ UNDER THE REDUCED NORM MAP. IN THIS NOTE WE INVESTIGATE THE GROUP G(D) BY TWO METHODS: BY USING VALUATION THEORY FOR DIVISION ALGEBRAS AND BY USING K -THEORY METHODS. THIS ENABLES US TO COMPUTE THE GROUP G(D) IN SEVERAL CASES. IN PARTICULAR, WE GIVE AN EXPLICIT FORMULA FOR G(D) WHEN F IS A LOCAL FILED AND D IS A TAME DIVISION ALGEBRA.