Fields of definition for admissible groups

TL;DR

This paper explores various notions of admissibility for finite groups over fields, analyzing how different definitions relate and establishing a comprehensive logical framework for these concepts.

Contribution

It provides a complete classification of the logical implications between nine variants of admissibility notions over fields.

Findings

Complete determination of implications between admissibility variants

Clarification of how definitions over subfields relate

Framework for understanding admissibility in division algebras

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.