2025/03/22 by Khanh, Huynh Viet, Khoa, Nguyen Duc Anh
#16K20 #16W60 #19B99 #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2503.17714
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 TK1(D) the torsion subgroup of the Whitehead group \rm K1(D) = D^*/D', where D^* is the multiplicative group of D and D' is its derived subgroup. Let \bf G be the subgroup of D^* such that TK1(D) = \bf G/D'. In this note, we prove that either (1 + MD) ∩ \bf G ⊆ D', or the residue field K has characteristic p > 0 and the group \bf H := ((1 + MD) ∩ \bf G)D'/D' is a p-group. Additionally, we provide examples of valued division algebras with non-trivial \bf H. This illustrates that, in contrast to the reduced Whitehead group \(\rm SK1(D)\), a complete analogue of the Congruence Theorem does not hold for \(\rm TK1(D)\).