We investigate the group valued functor G(D)=D^*/F^*D' where D is a division algebra with center F and D' the commutator subgroup of D^*. We show that G has the most important functorial properties of the reduced Whitehead group SK_1. We then establish a fundemental connection between this group, its residue version and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK_1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.