# Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It is the face lattice of the square polytope (see here), let's call this lattice $$P_4$$.

By some tools used in this paper and the computation of Gordon Royle cited here, we can prove that if $$P_4=[H,G]$$ as a lattice, then $$|G:H| \ge 135$$ (and so $$|G| \ge 270$$, because $$H \neq 1$$). Moreover, we checked (GAP) that $$|G| \ge 512$$, and for $$G$$ simple, $$|G| \ge |{\rm PSL}(2,191)|= 3483840$$.

The existence of a lattice which is not the lattice of an interval of finite groups is an open problem.
In the following paper (p72): Overgroup lattices in finite groups of Lie type containing a parabolic Michael Aschbacher conjectures (after John Shareshian) that a lattice $$L$$ such that the poset $$\overline{L}:=L \setminus \{\hat{0},\hat{1} \}$$ is disconnected with connected components the posets $$\overline{B}_{n_1}, \dots , \overline{B}_{n_r}$$, with $$B_{n_i}$$ a Boolean lattice of rank $$n_i \ge3$$ (and $$r \ge 2$$), is not the lattice of an interval of finite groups. But if all the $$n_i$$ are equal and odd, then $$L$$ is a non-Boolean Eulerian lattice (here the smallest example).

All these evidences lead to wonder:
Question: Is there a non-Boolean Eulerian interval of finite groups?

From an eventual positive answer for the relative version of K.S. Brown's problem and dual (i.e. $$\sum_{K \in [H,G]} \mu(K,G)|G:K|$$ and $$\sum_{K \in [H,G]} \mu(H,K)|K:H|$$ are nonzero) and the property here, we can deduce the following extension of (dual) Ore's theorem on Boolean intervals:

Conjecture: Let $$[H,G]$$ be an Eulerian interval of finite groups. Then:

• $$\exists V$$ irr. $$\mathbb{C}$$-rep. of $$G$$ such that $$G_{(V^H)} = H$$ (see here).
• $$\exists g \in G$$ such that $$\langle Hg \rangle = G$$ (see here).

This conjecture could help the investigation.

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