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Theorem 7.5 Let G be a group. (1) G has a unique identity element (2) Cancellation Laws. For all a, b,ce G, if ab ac, then b-c. For all a, b,c E G, if ba-ca, then (3) Each element of G has a unique inverse: For each a E G, there exists a unique element d e G such that ad-e and da e . Prove that each element of a finite group G appears exactly once in each row and exactly once in each column of the operation table. Hint: Suppose G-aaan), where a,aan are all distinct. What are the elements in the kth row of the operation table? Use the cancellation laws in Theorem 7.5(2).] a1.0....,a
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