Strengthen the subgroup-interval bound using the number of prime factors

Establish a bound of the form |Sub(G,H)| < n^{\frac14\Omega(n)+c} for an absolute constant c, where H\leq G are finite groups, n=[G:H]>1, and \Omega(n) denotes the number of prime factors of n counted with multiplicity.

Background

The paper proves the sharp bound |Sub(G,H)|<c(p)n^{\frac14\log_p n}, where p is the least prime divisor of the index n=[G:H]. This removes a previously known polynomial loss in the exponent. However, the authors explicitly note that their result does not establish a potentially stronger estimate whose exponent depends on \Omega(n), the total number of prime factors of n counted with multiplicity, rather than only on the least prime divisor. They identify this strengthening with a conjectural direction discussed in earlier work.