On maximal subgroups in division rings. (Q2566573)

This paper studies maximal subgroups of the multiplicative group of a division ring, in particular through conjectures raised by \textit{S. Akbari, M. Mahdavi-Hezavehi}, and \textit{M. G. Mahmudi} [J. Algebra 217, No. 2, 422-433 (1999; Zbl 0933.16033)]. Let \(D\) be a division ring with center \(F\), and \(M\) a maximal subgroup of \(D^\times\). The authors show that if \(D\) is algebraic over \(F\), then the subfield generated by the center of \(M\) contains \(F\), but has no intermediate subfields and no automorphisms over \(F\). If \(D\) is a quaternion algebra in characteristic not \(2\), then \(M\) cannot be nilpotent. Furthermore, it is shown that all the solvable maximal subgroups of the multiplicative group of the real quaternion algebra \(\mathbb{R}[i,j\mid i^2=j^2=(ij)^2=-1]\) are conjugate to \(\mathbb{C}^\times\cdot\langle j\rangle\).