Torsion subgroups of Whitehead groups of graded division algebras

In this paper, we study the torsion subgroup, which is denoted by ${\mathrm{T}\mathrm{K}}_1(E)$, of the Whitehead group $E^{\ast }\mathrm{/}[E^{\ast },E^{\ast }]$ of a graded division algebra $E$ which is finite dimensional over its center. In particular, we provide formulas for ${\mathrm{T}\mathrm{K}}_1(E)$ in some special cases such as when $E$ is unramified, totally ramified or semiramified. Using the formulas, we are able to compute the group ${\mathrm{T}\mathrm{K}}_1(D)$ of a valued division ring $D$, which is finite dimensional over a henselian center.