Breaking points in centralizer lattices

Abstract

In this note, we prove that the centralizer lattice C(G) of a group G cannot be written as a union of two proper intervals. In particular, it follows that C(G) has no breaking point. As an application, we show that the generalized quaternion 2-groups are not capable.MSC 2010 : Primary 20D30; Secondary 20D15, 20E15.