Embedding problems with local conditions and the admissibility of finite groups

TL;DR

This paper proves that certain embedding problems with local conditions over fields of positive characteristic are always solvable when the kernel is a p-group, and applies this to the admissibility of finite groups over global fields.

Contribution

It establishes proper solvability of embedding problems with local conditions for p-group kernels over fields of positive characteristic, and explores implications for group admissibility.

Findings

Every finite embedding problem with local conditions and p-group kernel is properly solvable.

Application to admissibility of finite groups over global fields of positive characteristic.

Provides an alternative proof for a known result of Sonn.

Abstract

Let $k$ be a field of characteristic $p>0$, which has infinitely many discrete valuations. We show that every finite embedding problem for $\Gal(k)$ with finitely many prescribed local conditions, whose kernel is a $p$-group, is properly solvable. We then apply this result in studying the admissibility of finite groups over global fields of positive characteristic. We also give another proof for a result of Sonn.