[Paper Review] Groups with large Noether bound

Kálmán Cziszter, M. Domokos|arXiv (Cornell University)|May 3, 2011

Finite Group Theory Research21 references3 citations

TL;DR

This paper classifies finite groups with a Noether number at least half their order, showing that — except for four sporadic exceptions — such groups are precisely those with a cyclic subgroup of index at most two. The authors use generalized Noether numbers and subgroup/subquotient reduction techniques to prove that the ratio β(G)/|G| ≥ 1/2 holds only for these groups or the exceptional cases: Z₃×Z₃, Z₂×Z₂×Z₂, A₄, and the binary tetrahedral group ŠA₄.

ABSTRACT

The finite groups having an indecomposable polynomial invariant whose degree is at least half of the order of the group are classified. Apart from four sporadic exceptions these are exactly the groups having a cyclic subgroup of index at most two.

Motivation & Objective

• To classify all finite groups G for which the Noether number β(G) is at least half the group order |G|.
• To determine the precise structure of groups achieving a large Noether bound, especially those with β(G)/|G| close to 1/2.
• To resolve the theoretical status of the 1/2 threshold in the Noether bound, showing it is sharp and only finitely many non-cyclic groups exceed it beyond this point.
• To extend the understanding of Noether numbers beyond known results, particularly for non-cyclic groups with cyclic subgroups of index 2.
• To provide a complete classification of groups with large Noether bound, including exact values and bounds for specific small groups.

Proposed method

• Introduce a generalized Noether number β_k(H) for subgroups H of index k in G, enabling estimates of β(G) via subgroup and quotient data.
• Use reduction lemmata (Lemma 1.2, Lemma 1.4) to relate β(G) to β(H) and β(G/N) for subgroups H and normal subquotients N.
• Apply known bounds on β(G) for specific groups: Z_p×Z_p, A₄, ŠA₄, and dihedral or semidirect products, using results from prior works.
• Leverage the fact that β(G) ≥ β(H) for subgroups H, and β(G)/|G| ≤ β(K)/|K| for subquotients K, to eliminate candidates.
• Use classification of minimal simple groups (from [40]) to rule out certain subquotients that would otherwise allow β(G)/|G| ≥ 1/2.
• Combine estimates from Propositions 1.13, 1.14, 3.1, 3.4, and Corollaries 1.8, 2.9, 3.2, 3.7 to bound β(G)/|G| in each case.

Experimental results

Research questions

• RQ1 Which finite groups G satisfy β(G) ≥ ½|G|, where β(G) is the Noether number?
• RQ2 Are there only finitely many non-cyclic groups with β(G)/|G| > 1/2, and what is the structure of those that achieve this?
• RQ3 What is the exact value or tight bound of the Noether number for groups like Z₃×Z₃, Z₂×Z₂×Z₂, A₄, and ŠA₄?
• RQ4 How does the generalized Noether number β_k(H) help in estimating β(G) from subgroup and quotient data?
• RQ5 Can the 1/2 threshold in the Noether bound be characterized algebraically and group-theoretically, and is it sharp?

Key findings

• The only finite groups G with β(G) ≥ ½|G| are those with a cyclic subgroup of index at most two, or the four exceptional groups: Z₃×Z₃, Z₂×Z₂×Z₂, A₄, and the binary tetrahedral group ŠA₄.
• For all non-cyclic groups with a cyclic subgroup of index 2, the Noether number satisfies β(G) − ½|G| ∈ {1, 2}, showing the bound is tight.
• The ratio β(G)/|G| < 1/2 for all other groups, and there are no limit points of this ratio in (1/2, 1), implying only finitely many groups exceed any c > 1/2.
• For G = Z₂×Z₂×Z₂, the Noether number satisfies β(G)/|G| ≤ 7/16 < 1/2, and similarly for other subquotients like D₂p×D₂q or (Z₂×Z₂)⋊Z₉.
• The group A₄ has β(A₄) = 4, so β(A₄)/|A₄| = 1/3, and for ŠA₄, β(ŠA₄) ≤ 12, so β(ŠA₄)/|ŠA₄| ≤ 1/2.
• The bound β(G)/|G| ≤ 1/4 + 3/(8k) holds for groups G with a subgroup H ≅ Z₂×Z₂×Z₂ of index k, and equality is only possible when k = 1.