By Jacques Dixmier

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If det(m) = n, with n an integer, then det(m −1 ) = 1/n. Since the determinant of any matrix composed of integers must be an integer, the condition is that det(m) = ±1. The subset of G L(2; Z) with determinant +1 forms the subgroup S L(2; Z) ⊂ G L(2; Z). The modular group P S L(2; Z) is obtained by identifying each pair of −b matrices in S L(2; Z) of the form [ −a ] [ ac db ]. 34) where F(n) is the nth Fibonacci number, defined recursively by F(n) = F(n − 1) + F(n − 2) n F(n) 0 0 1 1 2 1 3 2 4 3 5 5 6 8 7 ··· 13 · · · The proof by induction is simple.

This is not at all obvious, but will become clear when we discuss the infinitesimal properties of Lie groups in Chapter 4. The dimension of the manifold that parameterizes a Lie group is the dimension of the Lie group. It is the number of continuous real parameters required to describe each operation in the group uniquely. 2. Every matrix in S L(2; R) can be written as the product of a symmetric matrix and a rotation matrix, both unimodular. The symmetric matrix is parameterized by a two-dimensional manifold, the two-sheeted hyperboloid z 2 − x 2 − y 2 = 1.

Called group elements or group operations) together with a combinatorial operation ◦ (called group multiplication) form a group G if the following axioms are satisfied. (i) Closure: if gi ∈ G, g j ∈ G, then gi ◦ g j ∈ G. 1) where α, β, γ , δ are real numbers. This set forms a group under matrix multiplication. This is verified by checking that the group axioms are satisfied. (i) Closure if A and B are real 2 × 2 matrices, and A ◦ B = C (where ◦ now represents matrix multiplication), then C is a real 2 × 2 matrix.