Some finite groups which appear as Gal L/K, where \(K\subseteq {\mathbb{Q}}(\mu _ n)\)
The topic of this paper is the famous 'inverse problem' of Galois theory: Which groups occur as Galois groups over a fixed ground field k? \(k={\mathbb{Q}}\) or k a small extension of \({\mathbb{Q}}\) are of special interest. The most striking consequence of this work is the realization of the Fischer-Griess monster as a Galois group over \({\mathbb{Q}}.\) The author deals more general with non-abelian finite simple groups F that are 'rigid'. This means: there are conjugacy classes \(C_ 1,...,C_ k\) in F, such that i) F is generated by \(x_ 1,...,x_ k\) for every \((x_ 1,...,x_ k)\in C_ 1\times...\times C_ k\) with \(x_ 1\cdot...\cdot x_ k=1,\) and ii) F operates (by conjugation) transitively on these generators \((x_ 1,...,x_ k).\) This is the fundamental group theoretical concept. Examples of rigid simple groups given by the author are the Fischer-Griess monster and the groups \(PSL(2,2^ n)\). The author then constructs Galois extensions \(L| K\) of some cyclotomic field K with a given rigid group F as Galois group, and more general: Galois extensions \(N| {\mathbb{Q}}\) with a Galois group built out of F in an explicit form. The construction of these field extensions is based on the theory of Riemann surfaces and of discrete subgroups of \(PSL_ 2({\mathbb{R}})\). The proof is rather long and involved, containing a study of the congruence subgroup \(\Gamma_ 0(12)\) with its corresponding modular curve \(X_ 0(12)\), and explicit calculations in terms of Puiseux-series. To fix the ideas behind all this one should add: A group G generated by \(x_ 1,...,x_ k\) \(with\quad x_ 1\cdot...\cdot x_ k=1\) can be realized as the group of covering transformations of some compact Riemann surface \(X\to {\mathbb{P}}^ 1({\mathbb{C}})\), unramified outside k points in \({\mathbb{P}}^ 1({\mathbb{C}})\). Hence G is the Galois group of an algebraic function field L over the field of rational functions \({\mathbb{C}}(T)\). Here \({\mathbb{C}}\) may be replaced by \({\bar {\mathbb{Q}}}\), the field of algebraic numbers. The main problem then is, to reduce \({\bar {\mathbb{Q}}}\) further to some small subfield K. In the authors exposition \(X_ 0(12)\) plays the role of \({\mathbb{P}}^ 1({\mathbb{C}})\), and the ramification points are the six cusps of \(X_ 0(12)\). To reduce the field K of definition for \(L| {\mathbb{C}}(T)\) he uses explicit calculations in terms of Puiseux-series. One should add that in general - by the work of B. H. Matzat - one can get information about K in purely group theoretical terms of F, and at this point the rigidity of F comes in.
Fischer-Griess monster as Galois group over \({\mathbb{Q}}\)
discrete subgroups of \(PSL_ 2({\mathbb{R}})\)
scientific article; zbMATH DE number 3879038