# Lower bounds on the Noether number

## TL;DR

This paper demonstrates that the lower bounds for the Noether number, derived from subgroups and factor groups, are strictly less than the actual Noether number, by analyzing coinvariant algebras of induced representations.

## Contribution

It proves that inequalities used for lower bounds are strict for proper subgroups and factor groups, advancing understanding of Noether number bounds.

## Key findings

- Inequalities for Noether number bounds are strict for proper subgroups.
- Analysis of coinvariant algebras of induced representations supports the results.
- Provides new insights into the algebraic structure related to the Noether number.

## Abstract

The best known method to give a lower bound for the Noether number of a given finite group is to use the fact that it is greater than or equal to the Noether number of any of the subgroups or factor groups. The results of the present paper show in particular that these inequalities are strict for proper subgroups or factor groups. This is established by studying the algebra of coinvariants of a representation induced from a representation of a subgroup.