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 \(\hat{\varphi}\). For any Boolean group-complemented interval, we observe that \(\hat{\varphi} = \varphi \neq 0\) by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that \(\hat{\varphi}\) 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 \(\Delta(P)\) is \(\hat{\varphi}\), so nonzero. We deduce that these results are true beyond the group-complemented case with \(|G:H|