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On a Combinatorial Characterization of Abelian Group Actions
Abstract
Let G be a finite abelian group of order n and X a finite G-set. We give a necessary and sufficient condition on G for any X to be determined up to isomorphism by certain combinatorial data. Specifically, if MHX is the number of H-orbits in X, and adX is the number of G-orbits in X of cardinality d, we characterize when equalities MHX1 = MHX2 and adX1=adX2 for all H ≤ G and d|n, respectively, force an isomorphism X1 ≃ X2. Our main result is that this occurs if and only if G is cyclic or G is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to L-function factorization.
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BibTeX
@misc{gull2026combinatorial,
title = {{On a Combinatorial Characterization of Abelian Group Actions}},
author = {Carson Gull and Arthur Illidge and Benjamin Mudrak and Joshua Park and Shaver Phagan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16880},
eprint = {2610.16880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16880},
}
RIS
TY - UNPB
TI - On a Combinatorial Characterization of Abelian Group Actions
AU - Carson Gull
AU - Arthur Illidge
AU - Benjamin Mudrak
AU - Joshua Park
AU - Shaver Phagan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16880
UR - https://arxiv.org/abs/2610.16880
AB - Let G be a finite abelian group of order n and X a finite G-set. We give a necessary and sufficient condition on G for any X to be determined up to isomorphism by certain combinatorial data. Specifically, if M_H^X is the number of H-orbits in X, and a_d^X is the number of G-orbits in X of cardinality d, we characterize when equalities M_H^X_1 = M_H^X_2 and a_d^X_1=a_d^X_2 for all H ≤ G and d|n, respectively, force an isomorphism X_1 ≃ X_2. Our main result is that this occurs if and only if G is cyclic or G is the direct sum of a finite cyclic group and a cyclic group of prime order. We conclude with an arithmetic application to L-function factorization.
ER -