The finite quotients of the multiplicative group of a division algebra of degree \(3\) are solvable (Q1806271)

Let \(D\) be a division algebra of finite degree \(n\) over its centre \(F\). Given a normal subgroup \(N\) of \(D^*=D\setminus\{0\}\) containing \(F^*\) and \(H=D^*/N\), the authors show that the centre of \(H\) (and hence \(H/H'\)) has exponent dividing \(n\). They further prove that when \(n=3\), the finite quotients of \(D^*\) are soluble. The proof uses the factorization theorem of \textit{J. H. M. Wedderburn} [Trans. Am. Math. Soc. 22, 129-135 (1921; JFM 48.0126.01)], which states that the right zeros of \(fg\) are right zeros of \(g\) or conjugates of right zeros of \(f\).