On Boolean intervals of finite groups

TL;DR

This paper extends Ore's theorem to Boolean intervals in finite groups, showing that a dual Euler totient is nonzero in these cases and exploring implications in representation theory.

Contribution

It proves a dual version of Ore's theorem for Boolean group intervals, establishes the nonzero nature of the dual Euler totient, and links these concepts to Cohen-Macaulay properties.

Findings

Dual Euler totient is nonzero for Boolean group intervals.

The graded coset poset is Cohen-Macaulay for Boolean group-complemented intervals.

Results hold for intervals with index less than 32 and for Borel subgroups.

Abstract

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 $G$.