This randomized phase III trial studies how well crizotinib works and compares it to placebo in treating patients with stage IB-IIIA non-small cell lung cancer that has been removed by surgery and has a mutation in a protein called ALK. Mutations, or changes, in ALK can make it very active and important for tumor cell growth and progression. Tumors with this mutation may respond to treatments that target the mutation, such as crizotinib. Crizotinib may stop the growth of tumor cells by blocking the ALK protein from working. It is not yet known if crizotinib may be an effective treatment for treating non-small cell lung cancer with an ALK fusion mutation.