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The aim of this paper is to study properties of solutions to the fractional p-subLaplace equations on the Heisenberg group. Based on the maximum principles and the generalization of the direct method of moving planes, we obtain the symmetry and monotonicity of the solutions on the whole group and the Liouville property of solutions on a half space. MSC: Primary 35A01; secondary 35J57; 35D99doi:10.1186/s13661-020-01425-1 fatcat:l2k6kav6rraodajoibam7iqnxy