Multiplicative groups of division rings. (Q2923252)

This long article is basically a survey of the current state of our knowledge of the group structure of the multiplicative group \(D^*\) of a division ring \(D\). A substantial amount of the material is proved, often including alternative proofs and some improvements of known results, sometimes with shorter proofs yielding partial results. Having a homogenized account of all this material in one place is very valuable, especially for someone like the present reviewer who has been less than diligent in following the very substantial progress that has been made in this area in the last thirty years or so.NEWLINENEWLINE The paper is divided into nine sections and we briefly indicate the content of each of these.NEWLINENEWLINE 1. This summarizes our knowledge from Wedderburn (1905) to about 1970 and introduces the authors' notation. (One idiosyncrasy I noted; \(\zeta_iG\), one of the standard notations for the upper central series of a group \(G\), here denotes its lower central series, so \(\zeta_1G\) for example denotes its derived subgroup and not its centre.)NEWLINENEWLINE 2. This lengthy section studies the soluble-by-finite subgroups of \(D^*\) and contains a substantial amount of proof. (Page 42 contains the `Proof of Theorem 2'. Unfortunately the paper contains a Theorem 2.1, a Theorem 2.2, Theorem 2.4 etc., but no Theorem 2.)NEWLINENEWLINE 3. This section studies the finitely generated normal subgroups of the multiplicative group of a division ring and 4. studies its maximal subgroups. In particular it considers the existence or otherwise of non-Abelian free subgroups in a maximal subgroup of \(D^*\).NEWLINENEWLINE 5. This section considers the upper central series of \(D^*\). In the second half it introduces the notion of a valuation on a division ring and uses this to study the upper central series of \(D^*\), where for example, the centre \(F\) of \(D\) is a Henselian field and \(D\) is tame, unramified and finite-dimensional over \(F\).NEWLINENEWLINE 6. Here the \(K\)-theory, mainly the reduced \(K\)-theory of a division ring \(D\) finite-dimensional over its centre \(F\), is studied. Thus a particular group for study is \(SK_1(D)\), the kernel of the map of \(K_1(D)\) to \(K_1(F)\) induced by the reduced norm. Thus \(SK_1(D)\) is the subgroup of \(D^*\) of elements of reduced norm 1, taken modulo the derived subgroup \(\zeta_1D^*\) of \(D^*\).NEWLINENEWLINE 7. This section considers graded division algebras and in particular looks at the correspondence between the \(SK_1\) of a division algebra with a valuation and the \(SK_1\) of its associated graded division algebra.NEWLINENEWLINE 8. Here the authors' look at maximal subgroups of \(D^*\) and in particular consider the implications for \(D\) if \(D^*\) has no maximal subgroups. A tool here is the group \(G(D)\), defined to be \(D^*\) modulo the product of its derived subgroup and the image of \(D^*\) under the reduced norm (assuming \(D\) is finite-dimensional, etc.).NEWLINENEWLINE 9. This final section considers radicable division algebras.NEWLINENEWLINE The paper concludes with a very complete bibliography of some 106 items.