On Boolean intervals of finite groups

摘要

We prove a dual version of circle divide ystein Ore's theorem on distributive intervals in the subgroup lattice of finite groups, having a nonzero dual Euler totient phi. For any Boolean group-complemented interval, we observe that phi = phi not equal 0 by the original Ore's theorem. We also discuss some applications in representation theory. We conjecture that cp 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 P=C (H,G) is Cohen Macaulay and the nontrivial reduced Betti number of the order complex Delta(P) is phi, so nonzero. We deduce that these results are true beyond the group-complemented case with vertical bar G : H vertical bar 32. One observes that they are also true when H is a Borel subgroup of G. (C) 2018 Elsevier Inc. All rights reserved.