Permutations avoiding arithmetic patterns (Q1883652)

Let \(G\) be a group and \(\Omega_n(G)\) be the subset of all elements of \(G\) of order dividing \(n\). A permutation \(\pi\) is said to avoid a \(k\)-term arithmetic progression if there does not exist any \((a_1, \ldots , a_k)\), (not all \(a_i\) equal), such that for \(i=1, \ldots , k-2\), \(a_i+a_{i+2} = a_{i+1} +a_{i+1}\) and \(\pi(a_i)+\pi(a_{i+2}) = \pi(a_{i+1}) +\pi(a_{i+1})\). The author proves: 1) If \(G\) is an infinite abelian group, then there exists a permutation which avoids arithmetic progressions of length 3, if and only if the factor group \(G/\Omega_2(G)\) has the same cardinality as \(G\). He proves there are permutations of \(\mathbb{Z}_n\) avoiding 4-progressions, if and only if \(n \not= 2,3\), and conjectures there are permutations of \(\mathbb{Z}_n\) avoiding 3-progressions, if and only if \(n \not= 2,3,5,7\). 2) If \(G\) is a countably infinite abelian group, then there exists a Sidon set permutation of \(G\) if and only if the factor group \(G/\Omega_2(G)\) is infinite.