Sharp bounds for intervals in finite subgroup lattices
Let H be a subgroup of a finite group G, let n = [G:H] > 1, and let p be the least prime divisor of n. This Lean development proves an exact finite rational upper envelope for the number of subgroups K with H ≤ K ≤ G and the strict bound c(p) n^((log_p n)/4), where c(p) is an Euler-product times a Gaussian theta series. It formalizes the bandwise moment inequality that drives the proof, a universal base-two bound with an exact decimal coefficient, the exact optimal answer for prime-power indices, and elementary-abelian asymptotics showing that both the constant and the exponent 1/4 are…
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