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 $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) in G^3$ satisfying $b-a = c-b eq 0$. 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 $G = mathbb{Z}/nmathbb{Z}$ for all $n eq 2,3,5,7$, 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 $(mathbb{Z}/pmathbb{Z})^k$ if and only if $p$ is odd and $(p,k) otin {(3,1),(5,1), (7,1)}$.
تحميل البحث