Ph.D Thesis
Ph.D Student Danny Neftin Admissibility and Finite Groups over Number Fields Department of Mathematics Full Professor Sonn Jack

Abstract

A finite group G is called K-admissible over a field K if there exists a Galois extension L/K with Galois group G such that L is a maximal subfield of a division algebra with center K.

Over number fields K, Schacher presented a criterion for admissibility in the form of a realization problem (of groups as a Galois group) with prescribed local conditions. We extract these necessary local conditions and call them preadmissibility. It turns that often K-preadmissibility also implies K-admissibility. The thesis consists of three parts in which we study admissibility using preadmissibility.

In the first part, we compare admissibility to preadmissibility for various families of groups and show that often these two notions coincide. For abelian groups this comparison is closely related to the Grunwald-Wang theorem. We shall use the Grunwald-Wang theorem to show that there exist only very speical examples of K-preadmissible abelian groups A that are not K-admissible and give the precise conditions on A and K under which this phenomena occurs.

Over the field of rational numbers Q, it is conjectured that Q-preadmissibility implies Q-admissibility. This conjecture was proved for solvable groups by Sonn. In the second part, we generalize a theorem of Liedahl and Sonn's theorem to arbitrary number fields. We define the notion of tame admissibility which explains the kind of admissibility that occurs over Q and in this generalization.

In the third part, we discuss an arithmetic equivalence relation that is induced by admissibility. Namely, two number fields K and L are equivalent by admissibility if they have the same admissible groups. In 1985, Sonn asked whether two number fields K and L that are equivalent by admissibility necessarily have the same degree over Q. We provide evidence for a negative answer to this problem by constructing infinitely many pairs of number fields that have the same preadmissibile groups and the same odd order admissible groups.