Frattini subgroup of the unit group of central simple algebras. (Q2893001)

Let \(G=\mathrm{GL}(n,D)\), where \(n>1\) is a positive integer and \(D\) is a division ring of finite dimension over its centre \(F\). The authors are interested in calculating the Frattini subgroup \(\Phi(G)\) of \(G\). Their results depend very much on the structure of \(F\). For example, if \(F\) is a real global field then \(\Phi(G)\) is the product of \(\Phi(F^*)\) and the centre of the derived subgroup of \(G\). For other global fields \(\Phi(G)\) is equal to a specified subgroup of \(F^*\) containing \(\Phi(F^*)\) and depending only \(n\) and the dimension of \(D\) over \(F\). If \(F\) is a finitely generated extension of an algebraically closed field \(K\), then \(\Phi(G)=K^*\). Other examples are given in the paper.