The Noether number for the groups with a cyclic subgroup of index two

TL;DR

This paper determines the exact degree bound for polynomial invariants of certain finite groups with a cyclic subgroup of index two, showing the Noether number is roughly half the group order plus a small constant.

Contribution

It establishes the precise Noether number for finite non-cyclic groups with a cyclic subgroup of index two, filling a gap in invariant theory.

Findings

Noether number equals half the group order plus 1 or 2

Exact degree bounds for generators of polynomial invariants

Characterization of groups with cyclic subgroup of index two

Abstract

The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the order of the group plus 1 or 2.