Central Embedding Problems, the Arithmetic Lifting Property and Tame Extensions of Q

This paper focuses on the following two questions relating to the inverse Galois problem: (1) (Beckmann [2]) Is every finite Galois extension of Q the specialization of a Galois branched covering of P defined over Q and with the same Galois group? (2) (Birch [3]) Given a finite group G, is there a tamely ramified normal extension F/Q with Gal(F/Q) ∼ = G? We obtain affirmative answers to these questions for every finite central extension group G of some nonsolvable groups G. For both problems, the lifting step from G to G is done via central twist-type arguments and the corresponding results for abelian (kernel) groups. In Section 2, we introduce some notations and definitions and we quote known results that will be used later on. Black already noted in [5] that an affirmative answer to Beckmann’s question for all alternating groups follows from a construction of Mestre. The main result of Section 3 is that, for n = 4, 6, 7, this also holds for every finite central extension group G of An. Apart from Mestre’s result, another essential tool in our arguments is the abelian case of Beckmann’s problem solved in [2]. Dèbes proved in [9] that Beckmann’s question for abelian groups also admits an affirmative answer over every field (instead of Q). As a consequence, our main result is