On a Combinatorial Characterization of Abelian Group Actions

Carson Gull; Arthur Illidge; Benjamin Mudrak; Joshua Park; Shaver Phagan

paper

On a Combinatorial Characterization of Abelian Group Actions

arXiv:2610.16880

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 $M_{H}^{X}$ 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 $H \leq G$ 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.