The Boolean intervals of Chevalley type are strongly non group-complemented

TL;DR

This paper investigates the structure of Boolean intervals in the subgroup lattice of finite Chevalley groups, showing that certain lattice complements are not group complements, thus revealing new structural properties.

Contribution

It proves that in finite Chevalley groups, the lattice complement of an element in a Boolean interval is not a group complement, using Zsigmondy's theorem.

Findings

Boolean intervals in Chevalley groups are not group complemented.

Lattice complements in these intervals do not correspond to group complements.

The proof employs number-theoretic results like Zsigmondy's theorem.

Abstract

Let G be a finite Chevalley group and B a Borel subgroup. Then the interval [B,G] in L(G) is Boolean. We prove, using Zsigmondy's theorem, that for any element P in the open interval (B,G), its lattice-complement P^c is not a group-complement.