A characterization of Nested Groups in terms of conjugacy classes

TL;DR

This paper characterizes nested groups, where the centers of irreducible characters form a chain, by establishing a correspondence with a set of subgroups linked to conjugacy classes, providing a new structural insight.

Contribution

It introduces a novel subgroup characterization of nested groups based on conjugacy classes, linking character theory with subgroup chains.

Findings

Nested groups are characterized by a chain of associated subgroups.

A set of subgroups related to conjugacy classes fully determines nestedness.

The subgroup chain criterion offers a new perspective on group structure.

Abstract

A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these subgroups form a chain.