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The effect of a finite quadruple junction mobility on grain growth evolution has been studied by means of computer simulations. For this purpose a special three-dimensional grain assembly is proposed, which permits a steady-state motion of the grain boundaries and junctions of the system. It was found that the behavior of the system is determined by the dimensionless parameter Λqp, which is related to the quadruple junction mobility. Numerous simulations were carried out in order to determine the effect of this parameter on grain growth. The results show that a finite quadruple junction mobility can slow down grain growth. However, the simulations also demonstrated that a finite triple line mobility hinders grain growth even more effectively.