On the non-triviality of \(G(D)\) and the existence of maximal subgroups of \(\text{GL}_1(D)\).
Let \(F\) be a field, \(D\) a central division \(F\)-algebra of Schur index \(\text{ind}(D)=n\), \(D^*\) the multiplicative group of \(D\), and \(D'\) the commutator subgroup of \(D^*\). Denote by \(G_0(D)\) and \(G(D)\) the quotient groups \(D^*/F^*D'\) and \(D^*/D'RN_{D/F}\), respectively, where \(RN_{D/K}\subseteq D^*\) is the group of reduced norms of \(D/K\). The paper under review proves that \(G_0(D)=\{1\}\), provided that \(D\) is not a quaternion algebra. It shows that if \(F\) contains a primitive \(p\)-th root of unity or \(\text{char}(F)=p\), for a given prime \(p\), and \(\text{ind}(D)\) is a power of \(p\), then \(G(D)\neq\{1\}\) unless \(D\) is a quaternion algebra. As explained by the authors, the motivation for this research comes from the generally open problem of whether \(D^*\) possesses a maximal subgroup with respect to inclusion.
scientific article; zbMATH DE number 2153341
scientific article