The Noether number of the non-abelian group of order 3p

TL;DR

This paper establishes degree bounds for polynomial invariants of a specific non-abelian group of order 3p, providing explicit generation and separating system bounds over fields of characteristic zero and coprime to 3p.

Contribution

It proves that the algebra of polynomial invariants is generated by elements of degree at most p+2 for the non-abelian group of order 3p and determines the exact degree bounds for separating systems.

Findings

Invariants generated by elements of degree ≤ p+2

Exact degree bounds for separating systems

Results hold over fields of characteristic zero and not dividing 3p

Abstract

It is proven that for any representation over a field of characteristic 0 of the non-abelian semidirect product of a cyclic group of prime order p and the group of order 3 the corresponding algebra of polynomial invariants is generated by elements of degree at most p+2. We also determine the exact degree bound for any separating system of the polynomial invariants of any representation of this group in characteristic not dividing 3p.