The K-admissibility of SL(2,5)
If L/K is a finite field extension, then L is called K-adequate if L is a maximal subfield of a division algebra with center K. A finite group G is K-admissible if G is isomorphic to the Galois group of an adequate Galois extension L/K. A group H is called a double cover of a group G if there exists a nonsplit exact sequence \(1\to Z\to H\to G\to 1\) (cardinality of Z is 2). If \(n\geq 4\), \(A_ n\) (alternating group) has one (up to isomorphism) double cover; also if \(n\geq 4\) and \(n\neq 6\), \(S_ n\) (symmetric group) has exactly two non-isomorphic double covers. The present paper restricts attention mainly to \(A_ 5\) (\(\simeq PSL(2,5))\) and \(S_ 5\) (\(\simeq PGL(2,5)).\) The main theorem, based on work of J. F. Mestre, establishes the equivalence of (1) and (2): (1) H is not K-admissible; (2) \(\sqrt{-1}\) is in K and K contains only one prime divisor of 2; where K is an algebraic number field, H a subgroup of SL(2,5) which contains an \(S_ 2\)-group T of SL(2,5). The same work can be used to give simple proofs for two known results: (a) If H is a subgroup of \(A_ 5\), containing an \(S_ 2\)-group T of \(A_ 5\), then H is K-admissible for every number field K; (b) If G is a group whose Sylow groups are metacyclic and such that all composition factors of G are either cyclic or isomorphic to \(A_ 5\), then G is \({\mathbb{Q}}\)-admissible.
scientific article; zbMATH DE number 4150336