  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 $\pi(b) - \pi(a) \neq \pi(c) - \pi(b)$
for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. 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 $n \neq 2,3,5,7$, 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 $(p,k) \not\in
\{(3,1),(5,1), (7,1)\}$.
