Breaking points in centralizer lattices

TL;DR

This paper proves that the centralizer lattice of a group cannot be decomposed into two proper intervals, implying it has no breaking points, and applies this to show certain groups are not capable.

Contribution

It establishes a structural property of centralizer lattices and applies it to classify generalized quaternion 2-groups as non-capable.

Findings

Centralizer lattice cannot be expressed as a union of two proper intervals.

Centralizer lattice has no breaking points.

Generalized quaternion 2-groups are not capable.

Abstract

In this note, we prove that the centralizer lattice ${\mathfrak C}(G)$ of a group $G$ cannot be written as a union of two proper intervals. In particular, it follows that ${\mathfrak C}(G)$ has no breaking point. As an application, we show that the generalized quaternion $2$-groups are not capable.