{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830003,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $n &gt; 1.4 \\times 10^{14}$. 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 $(p,k) \\not\\in \\{(3,1),(5,1), (7,1)\\}$.<\/jats:p>","DOI":"10.37236\/6379","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T14:26:26Z","timestamp":1578666386000},"source":"Crossref","is-referenced-by-count":0,"title":["Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups"],"prefix":"10.37236","volume":"24","author":[{"given":"Noam D.","family":"Elkies","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ashvin A.","family":"Swaminathan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2017,2,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p20\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p20\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T00:05:39Z","timestamp":1579219539000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/6379","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,2,3]]},"article-number":"P1.20"}}