On Boolean intervals of finite groups

Abstract

We prove a dual version of Øystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient $\hatφ$. For any Boolean group-complemented interval, we observe that $\hatφ = φ\neq 0$ by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that $\hatφ$ is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval $[H,G]$, the graded coset poset $\hat{P} = \hat{C}(H,G)$ is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex $Δ(P)$ is $\hatφ$, so nonzero. We deduce that these results are true beyond the group-complemented case with $|G:H|<32$. One observes that they are also true when $H$ is a Borel subgroup of $G$.