Stabilizing the axion and a natural solution to the mu problem of the supersymmetry

The axion solution to the strong CP problem makes use of a global Peccei-Quinn (PQ) U(1) symmetry which is susceptible to violations from quantum gravitational effects. We show explicitly how discrete gauge symmetries can protect the axion from such violations. PQ symmetry emerges as an accidental global symmetry from discrete gauge symmetries which are subgroups of the anomalous U(1) of string origin. We show how the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion model also provides a natural solution to mu problem of supersymmetry as mu ~ M_{SUSY} ~ M^2_{PQ}/M_{Pl}