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Axion models have a spontaneously broken Z(N) symmetry. The resulting discretely degenerate vacua and domain-wall solitons are incompatible with the standard cosmology. It is possible, however, to introduce a small Z(N) breaking interaction into axion models without upsetting the Peccei-Quinn mechanism. In that case the domain walls disappear a certain time after their formation in the early universe. Their presence for a limited time period might lead to galaxy formation.PACS numbers: 11.15. Ex, 11.30.Er, 12.10.En, 98.80.Bp When a gauge interaction explicitly breaks a global symmetry, it often happens that a discrete subgroup of the global symmetry remains unbroken. Such is the case in axion models 1 which, as I am about to show, have a spontaneously broken Z(N) symmetry. The result applies to all models which have a Peccei-Quinn symmetry UPQ(1) which is broken only by the QCD gluon anomaly. N is the number of quark flavors that rotate under UPQ(1). TO be specific however, I analyze the Dine-Fischler-Srednicki model 2 in which the axion can be made "invisible." The Yukawa couplings and scalar selfinteractions of that model, N/2

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