Permutations destroying arithmetic structure (Q2341051)

Summary: Given a linear form \(C_1X_1 + \cdots + C_nX_n\), with \(C_1,\dots,C_n\in \mathbb Z\), we characterize exactly the countably infinite abelian groups \(G\) for which there exists a permutation \(f\) that maps all solutions \((\alpha_1, \ldots, \alpha_n) \in G^n\) (with the \(\alpha_i\) not all equal) to the equation \(C_1X_1 + \cdots + C_nX_n = 0 \) to non-solutions. This generalises a result of \textit{P. Hegarty} [Electron. J. Comb. 11, No. 1, Research paper R39, 21 p. (2004; Zbl 1128.11007)] about permutations of an abelian group avoiding arithmetic progressions. We also study the finite version of a problem suggested by Hegarty [loc. cit.]. We show that the number of permutations of \(\mathbb{Z}/p\mathbb{Z}\) that map all 4-term arithmetic progressions to non-progressions, is asymptotically \(e^{-1}p!\).