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We consider the fine grading of s l ( n , C ) induced by tensor product of generalized Pauli matrices in the paper. Regard P G L ( n , C ) as the inner automorphism group of s l ( n , C ) . Based on the classification of maximal abelian subgroups of P G L ( n , C ) consisting of diagonalizable automorphisms by Havlicek et al., we prove that any such subgroup K of P G L ( n , C ) is a symplectic abelian group and its Weyl group, which describes the symmetry of the fine grading induced by the action of K, is just the isometry group of the symplectic abelian group K. For a finite symplectic abelian group, it is also proved that its isometry group is always generated by the transvections contained in it.
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