On the existence of normal maximal subgroups in division rings (Q1612114)

Let \(F\) be a field, \(D\) an associative central division algebra over \(F\), \(D'\) the commutator subgroup of the multiplicative group \(D^*\), and \(G(D)\) the quotient group of \(D^*\) by the subgroup \(F^*\cdot D'\). The paper under review provides sufficient conditions for the existence of a maximal subgroup of \(D^*\). It shows that if the \(p\)-component \(\text{Br}(F)_p\) of the Brauer group \(\text{Br}(F)\) is nontrivial, for some prime number \(p\), then \(F^*\) possesses a subgroup of index \(p\) in each of the following special cases: (i) \(F\) is of characteristic zero; (ii) \(F\) is of characteristic \(p\) or \(F\) contains a primitive \(p\)-th root of unity. Also, the authors prove that \(D^*\) possesses a maximal subgroup, provided that \(D\) is of finite dimension over \(F\) (or more generally, \(D\) is locally finite dimensional over \(F\), and there exists a positive integer \(m\) for which \(G(D)\) does not contain an element of order \(m\)). They show that if \(M\) is a maximal subgroup of \(D^*\) not containing \(F^*\), then its centre is a maximal subgroup of \(F^*\), and conversely, that if \(D\) is finite dimensional over \(F\) and \(M\) is a maximal subgroup of \(F^*\) including the centre of \(D^*\), then \(D\) possesses a maximal subgroup that is normal in \(D^*\) and contains \(M\). On the other hand, the paper shows that the multiplicative group of the division algebra of real (Hamiltonian) quaternions does not possess such a maximal subgroup. Finally, the paper gives a sufficient condition for the inclusion of each element \(\xi\in D^*\) in a normal maximal subgroup \(M_\xi\) of \(D^*\).