On Boolean intervals of finite groups
Mamta Balodi, Sebastien Palcoux

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 . For any Boolean group-complemented interval, we observe that by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that is always nonzero for Boolean intervals. In order to investigate it, we prove that for any Boolean group-complemented interval , the graded coset poset is Cohen-Macaulay and the nontrivial reduced Betti number of the order complex is , so nonzero. We deduce that these results are true beyond the group-complemented case with . One observes that they are also true when is a Borel subgroup of .
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