On Posets of Classes of Automorphic Subgroups of Finite Groups

TL;DR

This paper investigates the structure of classes of automorphic subgroups in finite groups, introducing a partial order, and characterizes when these classes form chains or distributive lattices.

Contribution

It introduces a new partial order on classes of automorphic subgroups and characterizes finite groups with chain or distributive lattice structures in this poset.

Findings

AutCl(Dn) and AutCl(Q4m) are distributive lattices

Characterization of groups where AutCl(G) is a chain

Addresses a previously posed open problem

Abstract

In [16], Tarnauceanu studied the poset Iso(G), of isomorphic classes of subgroups of a finite group G and proposed several questions for further research. In this paper, we study the poset AutCl(G), of classes of automorphic subgroups of finite group G. We introduce a partial order on AutCl(G) to tackle problem 5 mentioned in {\S}4 of [16]. More precisely, we prove that AutCl(Dn) and AutCl(Q4m) are distributive lattices. Moreover, we characterize all classes of finite groups for which AutCl(G) is a chain.