Ph.D Thesis | |

Ph.D Student | Neftin Danny |
---|---|

Subject | Admissibility and Finite Groups over Number Fields |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Jack Sonn |

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.