  We prove a dual version of {\O}ystein 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
$|G:H|<32$. One observes that they are also true when $H$ is a Borel subgroup
of $G$.
