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Given a linear form C 1 X 1 + · · · + C n X n , with coefficients in the integers, we characterize exactly the countably infinite abelian groups G for which there exists a permutation f that maps all solutions (α 1 ,. .. , α n) ∈ G n (with the α i not all equal) to the equation C 1 X 1 + · · · + C n X n = 0 to non-solutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arithmetic progressions. We also study the finite version of the problem suggested byfatcat:zr25iutv7zeq5if63gl5kdoqzi