Algebraic Geometry Seminar

Semiabelian Groups and the Inverse Galois Problem

Speaker: Benjamin Blum-Smith, Courant Institute of Mathematical Sciences

Location: Warren Weaver Hall 201

Date: Tuesday, April 1, 2014, 3:30 p.m.


The Inverse Galois Problem asks, for a given field K, what finite groups G can be realized as Galois groups of extensions of K. The problem was posed for K=Q by Hilbert, who also made the first significant breakthrough, showing that if G has a Galois realization over Q(x), it has one over Q. Thus K=Q(x) has also drawn a great deal of attention. If a field extension M/k(x) has Galois group G and additionally k is algebraically closed in M, M is said to be a *geometric* or *regular* realization of G over k. Geometric realizations have been particularly sought after because they automatically yield realizations over k'(x) for arbitrary extensions k' of k. It was discovered by Saltman (1982) and recently rediscovered by Bogomolov that the class of finite groups G with regular realizations over Q (and actually over any field) is closed under taking semidirect products with abelian kernel. Since it is also closed under quotients, this leads to the definition (Thompson, 1986) of *semiabelian group*, which is the smallest class of finite groups with both of these closure properties. Thompson found that all groups of nilpotency class 2 are semiabelian. Bogomolov asked recently whether this result could be extended to metabelian groups. We will present these results, and an example due to Dentzer showing that a metabelian group can fail to be semiabelian.