Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups

Abstract

Given an abelian group $G$, it is natural to ask whether there exists a permutation $\pi$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $\pi (b)-\pi (a)\ne \pi (c)-\pi (b)$ for every ordered triple $(a,b,c)\in G^3$ satisfying $b-a=c-b\ne 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/{\Omega }_2(G)$ has the same cardinality as $G$, where ${\Omega }_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G=\mathbb{Z}/n\mathbb{Z}$ for all $n\ne 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n>1.4\times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(\mathbb{Z}/p\mathbb{Z}{)}^k$ if and only if $p$ is odd and $(p,k)\notin \{(3,1),(5,1),(7,1)\}$.

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