Fields of definition for admissible groups

Abstract

A finite group G is admissible over a field M if there is a division algebra whose center is M with a maximal subfield G-Galois over M. We consider nine possible notions of being admissible over M with respect to a subfield K of M, where the division algebra, the maximal subfield or the Galois group are asserted to be defined over K. We completely determine the logical implications between all variants.