Breaking points in the poset of conjugacy classes of subgroups of a finite group

TL;DR

This paper characterizes finite groups with a specific property in their subgroup conjugacy class posets, revealing a new way to identify generalized quaternion 2-groups and exploring related generalizations.

Contribution

It provides a complete characterization of groups with breaking points in their subgroup conjugacy class posets, including a new characterization of generalized quaternion 2-groups.

Findings

Finite groups with breaking points are characterized.

A new characterization of generalized quaternion 2-groups is provided.

The property is generalized to broader classes of groups.

Abstract

In this note, we determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also studied.