On Boolean intervals of finite groups

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 φˆ. For any Boolean group-complemented interval, we observe that φˆ=φ≠0 by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that φˆ 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 Pˆ=Cˆ(H,G) is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex Δ(P) is φˆ, 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.