The normalizer of Γ₀(N): Newman's theorem, and the order and presentations of B(N)
Formal statements and proofs, in Lean 4 with Mathlib, about B(N) = Nor(Γ₀(N))/Γ₀(N), the normalizer of Γ₀(N) in SL₂(ℝ) modulo Γ₀(N), for every N ≥ 1: Newman's theorem describing the normalizer, in both directions; the order |B(N)| = 2^ω(n) ψ(N)/ψ(n), where h is the largest divisor of 24 with h² ∣ N and n = N/h²; generators and complete presentations of B(N) for every N; a criterion for B(N) to be the internal direct product of its local factors; B(18) ≅ S₄; and the errors in the published presentations: the Akbaş–Singerman relation fails in B(256), and Bars's presentations at levels 32, 64…
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