A crystal plasticity finite element method (CPFEM) model has been developed to investigate the effect of the coefficient of friction (COF) on the evolution of crystallographic texture and mechanical behaviour of the initially oriented aluminium single crystal during nano-indentation. The simulation results have been validated by comparing with the experimental observations. The load-displacement curves from the simulation with different COFs have been analysed and the calculated values of Young's moduli agree well with the real material. The indentation hardness with different COFs has been investigated and the deviation is quite small. The piling-up curve was captured on the deformed surface and it decreased when the COF increased. Sinking-in curve was also captured and it increased when the COF increased. The pole figures and the lattice rotation angles after unloading were calculated and then analysed. The analysis revealed that these lattice rotation angles in the chosen deformed zone were significantly affected by the COF during nano-indentation.