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 π of G that “destroys” all nontrivial 3-term arithmetic progressions (APs), in the sense that π(b)− π(a) 6= π(c)− π(b) for every ordered triple (a, b, c) ∈ G3 satisfying b − a = c − b 6= 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/Ω2(G) has the same cardinality as G, where Ω2(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 APdestroying permutation of G exists if G = Z/nZ for all n 6= 2, 3, 5, 7, and together with Martinsson, he has proven the conjecture for all n > 1.4 × 1014. 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 (Z/pZ)k if and only if p is odd and (p, k) 6∈ {(3, 1), (5, 1), (7, 1)}.

Cite this paper

@article{Elkies2017PermutationsTD, title={Permutations that Destroy Arithmetic Progressions in Elementary p-Groups}, author={Noam D. Elkies and Ashvin A. Swaminathan}, journal={Electr. J. Comb.}, year={2017}, volume={24}, pages={P1.20} }