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Moreover we construct for any n\\geq 2 an abelian semigroup G of GL(n;C) generated by n + 1 diagonal matrices which is locally hypercyclic but not hypercyclic and such that JG(e_k) = C^n for every k = 1; : : : ; n, where (e_1; : : : ; e_n) is the canonical basis of C^n. 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