The Noether numbers and the Davenport constants of the groups of order less than 32

Abstract

The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.

accepted in J. Algebra

References in corpus (3)

• The interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics
• The Noether number of the non-abelian group of order 3p
• The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup

Cited by in corpus (10)

• Erdős-Ginzburg-Ziv theorem and Noether number for $C_m{⋉}_{\phi }{C}_{mn}$
• On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II
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• On the invariant E(G) for groups of odd order
• On the Noether number of $p$-groups
• On an inverse problem of Erd\H os, Kleitman, and Lemke
• On zero-sum free sequences contained in random subsets of finite cyclic groups
• The $\{1,s\}$-weighted Davenport constant in $C_n^k$
• A reciprocity on finite abelian groups involving zero-sum sequences
• On product-one sequences over dihedral groups