Sharp bounds for intervals in finite subgroup lattices

Abstract: Let H be a subgroup of a finite group G, and put n = [G:H] > 1. If p is the least prime divisor of n, we prove that the number of subgroups K with H <= K <= G is less than c(p) n^(log_p n/4). Here c(p) is the product of (1 - p^-j)^-1 over all positive integers j, multiplied by the sum of p^-z^2 over all integers z. The exponent and the constant are simultaneously optimal, already for elementary abelian p-groups of even rank. In particular, for arbitrary n > 1, the number of intermediate subgroups is less than 7.371968802 n^(log_2 n/4). This removes the polynomial factor from the best previously known relative bound and extends the sharp absolute subgroup-counting theorem to arbitrary subgroup intervals. The group-theoretic argument is elementary. Its main ingredient is a weighted packing inequality for relative generating tuples, obtained from an escape tree on the coset space G/H.