On the Noether number of $p$-groups

TL;DR

This paper investigates the Noether number of p-groups, establishing conditions under which groups have large indecomposable polynomial invariants, and characterizes specific group structures related to this property.

Contribution

It provides a precise characterization of p-groups with large indecomposable invariants, linking group structure to invariant degrees.

Findings

Groups of order p^n have high-degree invariants if they contain a cyclic subgroup of index at most p.

Only specific small groups or those with certain cyclic subgroups achieve the maximal invariant degree.

The paper identifies exact structural conditions for the existence of large indecomposable invariants.

Abstract

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small order.