On ramification structures for finite nilpotent groups

TL;DR

This paper extends the understanding of ramification structures from abelian groups to a broader class of finite nilpotent groups, correcting previous errors and characterizing groups with specific Sylow p-subgroup properties.

Contribution

It generalizes the characterization of ramification structures to finite nilpotent groups with certain Sylow p-subgroups, including regular, powerful, and p-central p-groups, and corrects earlier inaccuracies.

Findings

Extended characterization to nilpotent groups with specific Sylow p-subgroups.

Corrected errors in previous results on abelian 2-groups.

Clarified the relation between ramification structures of groups and their Sylow 2-subgroups.

Abstract

We extend the characterization of abelian groups with ramification structures given by Garion and Penegini to finite nilpotent groups whose Sylow $p$-subgroups have a `nice power structure', including regular $p$-groups, powerful $p$-groups and (generalized) $p$-central $p$-groups. We also correct two errors in the result of Garion and Penegini regarding abelian $2$-groups with ramification structures and the relation between the sizes of ramification structures for an abelian group and those for its Sylow $2$-subgroup.