OEIS A000056: Order of SL2(Z/nZ)

Join us as we unpack SL2(Z/nZ), the group of 2×2 integer matrices modulo n with determinant 1, and its order—the number of distinct elements. We explore the surprising strong divisibility pattern of these orders, how to compute them via the Jordan function J2(n) and related multiplicative functions, and what this tells us about the internal structure of the group. We’ll connect these ideas to PSL(2, Z), the modular group acting on the upper half-plane, and see how the algebraic data reflects geometry through hyperbolic tessellations, modular forms, and real-world applications in cryptography and theoretical physics.

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