On the 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 \(\textrm{TK}_1(D)\) the torsion subgroup of the Whitehead group \(\textrm{K}_1(D) = D^*/D'\), where \(D^*\) is the multiplicative group of D and \(D'\) is its derived subgroup. Let \(\textbf{G}\) be the subgroup of \(D^*\) such that \(\textrm{TK}_1(D) = \textbf{G}/D'\). In this note, we prove that either \((1 + M_D) \cap \textbf{G} \subseteq D'\), or the residue field \(\overline{K}\) has characteristic \(p > 0\) and the group \(\textbf{H}:= ((1 + M_D) \cap \textbf{G})D'/D'\) is a p-group. Additionally, we provide examples of valued division algebras with non-trivial \(\textbf{H}\). This illustrates that, in contrast to the reduced Whitehead group \(\textrm{SK}_1(D)\), a complete analogue of the congruence theorem does not hold for \(\textrm{TK}_1(D)\).