A twelve-term exclusion for the small Davenport constant of E2× C3r
Abstract
Let E2 be the extraspecial group of order 35 and exponent three. For every r≥1, we prove that a product-one-free sequence of length 2r+11 over E2× C3r cannot contain exactly 2r−1 central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over 𝔽3. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of d(E2× C3r) for arbitrary r is not determined by this result.
Related
BibTeX
@misc{volkmann2026twelveterm,
title = {{A twelve-term exclusion for the small Davenport constant of $E_2\times C_3^r$}},
author = {Andreas Volkmann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22972},
eprint = {2609.22972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22972},
}
RIS
TY - UNPB
TI - A twelve-term exclusion for the small Davenport constant of E_2× C_3^r
AU - Andreas Volkmann
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22972
UR - https://arxiv.org/abs/2609.22972
AB - Let E_2 be the extraspecial group of order 3^5 and exponent three. For every r≥1, we prove that a product-one-free sequence of length 2r+11 over E_2× C_3^r cannot contain exactly 2r−1 central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over 𝔽_3. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of d(E_2× C_3^r) for arbitrary r is not determined by this result.
ER -