The admissibility of \(M_{11}\) over number fields
Let \(k\) be a field, \(D\) a central division \(k\)-algebra and \(G\) a finite group. We say that \(D\) is a \(G\)-crossed product if it admits a maximal subfield \(L\) that is a Galois extension of \(k\) with a Galois group \(\mathcal{G}(L/k)\) isomorphic to \(G\). A Galois extension \(L/k\) with \(\mathcal{G}(L/k) \cong G\) is called \(k\)-adequate, if \(L\) is a maximal subfield of a \(G\)-crossed product central division \(k\)-algebra. The problem of finding the \(k\)-admissible finite groups, i.e. those groups \(G\) for which there exists a \(G\)-crossed product division algebra central over \(k\), is a classical problem in the theory of central division algebras over special fields. This problem originates from the study of \(\mathbb{Q}\)-admissibility initiated by Schacher who proved that every \(\mathbb{Q}\)-admissible group has metacyclic Sylow subgroups (see [\textit{M. Schacher}, J. Algebra 9, 451--477 (1968; Zbl 0174.34103)]. Conversely, it has been proved by \textit{J. Sonn} [J. Algebra 84, 411--419 (1983; Zbl 0524.16007)] that finite solvable groups whose Sylow subgroups are metacyclic are necessarily \(\mathbb{Q}\)-admissible. Moreover, by Sonn's theorem, every group \(H\) of the considered type is strongly \(\mathbb{Q}\)-admissible, in the sense that, for each \(n \in \mathbb{N}\), there is a \(\mathbb{Q}\)-adequate Galois extension \(L/\mathbb{Q}\) with \(\mathcal{G}(L/\mathbb{Q}) \cong H\) and \(L \cap \mathbb{Q}(\mu_n) = \mathbb{Q}\), where \(\mu_n\) denotes the group of \(n\)-th roots of unity in a separable closure of \(\mathbb{Q}\). Generally, the question of whether finite groups with metacyclic Sylow subgroups are \(\mathbb{Q}\)-admissible remains open. It is known that the answer to this question reduces to proving the strong \(\mathbb{Q}\)-admissibility of the groups in an explicit list of five families (see [\textit{D. Chillag} and \textit{J. Sonn}, Isr. J. Math. 40, 307--323 (1981; Zbl 0496.20027)]. The last family in this list is the Mathieu group \(M_{11}\), the only sporadic group whose Sylow subgroups are metacyclic. The paper under review proves that \(M_{11}\) is \(\mathbb{Q}\)-admissible. Its main result states that if \(k\) is a number field in which at least one of the primes \(2\) and \(11\) splits completely, then \(M_{11}\) is infinitely often \(k\)-admissible (i.e. there exist infinitely many linearly disjoint adequate Galois extensions \(L/k\) with \(\mathcal{G}(L/k) \cong G\)). The idea of the proof of this theorem is to make it possible to apply Schacher's criterion, according to which the admissibility of \(G = M_{11}\) reduces to realizing \(G\) infinitely often as a Galois group of an extension \(L/k\) such that the noncyclic Sylow subgroups of \(M_{11}\) (the semidihedral group \(\text{SD}_{16} \cong C_8 \rtimes C_2\) and the direct product \(C_3 \times C_3\)) are contained in a decomposition group over at least two primes of \(k\). In order to implement this idea, the authors apply Hilbert's irreducibility theorem, weak approximation, and a specialization result of Beckmann (see Theorem~1.2 of: \textit{S. Beckmann} [J. Reine Angew. Math. 419, 27--53 (1991; Zbl 0721.11052)] which connects the inertia group of primes of \(\mathbb{Q}(t)\) in a regular extension \(E/\mathbb{Q}(t)\) with inertia groups of primes of \(\mathbb{Q}\) in a specialization \(E_a/\mathbb{Q}\), for \(t =a\).
scientific article; zbMATH DE number 6865843
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