Let G = GL(2, R), the group of 2 × 2 matrices over R with nonzero determinant. Let H be the subgroup of matrices of determinant ±1. If a, b ∈ G and aH = bH, what can be said about det (a) and det (b)? Prove or disprove the converse. [Determinants have the property that det (xy) = det (x)det (y).]
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