A transformation x' = A x is unimodular if the determinant of the matrix A satisfies det(A) = ± 1. A necessary and sufficient condition that a linear transformation transform a lattice to itself is that the transformation be unimodular. If z is a complex number, then the transformation z' = (a z + b)/(c z + d) is called a unimodular if a, b, c, and d are integers with a d - b c = 1. The set of all unimodular transformations forms a group called the modular group.
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