On Congruence Theorem for valued division algebras

Let $K$ be a field equipped with a Henselian valuation, and let $D$ be a tame central division algebra over the field $K$. Denote by ${\mathrm{T}\mathrm{K}}_1(D)$ the torsion subgroup of the Whitehead group ${\mathrm{K}}_1(D)=D^{\ast }\mathrm{/}{D}^{\mathrm{\prime }}$, where $D^{\ast }$ is the multiplicative group of $D$ and $D^{\mathrm{\prime }}$ is its derived subgroup. Let $\mathbf{G}$ be the subgroup of $D^{\ast }$ such that ${\mathrm{T}\mathrm{K}}_1(D)=\mathbf{G}\mathrm{/}{D}^{\mathrm{\prime }}$. In this note, we prove that either $(1+M_D)\cap \mathbf{G}\subseteq D^{\mathrm{\prime }}$, or the residue field $\bar{K}$ has characteristic $p>0$ and the group $\mathbf{H}:=((1+M_D)\cap \mathbf{G})D^{\mathrm{\prime }}\mathrm{/}{D}^{\mathrm{\prime }}$ is a $p$-group. Additionally, we provide examples of valued division algebras with non-trivial $\mathbf{H}$. This illustrates that, in contrast to the reduced Whitehead group ${\mathrm{S}\mathrm{K}}_1(D)$, a complete analogue of the Congruence Theorem does not hold for ${\mathrm{T}\mathrm{K}}_1(D)$.