On the minimal ramification problem for $\ell$-groups

TL;DR

This paper investigates the minimal ramification problem for finite p-groups, proving it for a specific family of groups including cyclic and Sylow p-subgroups, but not for all p-groups.

Contribution

It establishes the minimal ramification realization for a broad family of p-groups, expanding understanding of Galois groups with limited ramification.

Findings

Proved minimal ramification for a family of p-groups including cyclic and Sylow p-subgroups.

The family is closed under direct and wreath products, and rank-preserving homomorphic images.

Not all finite p-groups are included in this family.

Abstract

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups.