On Posets of Classes of Subgroups with Same Set of Orders of Elements

TL;DR

This paper investigates the structure of posets formed by classes of subgroups sharing the same element orders in finite groups, revealing conditions under which these posets are chains, lattices, or possess specific properties.

Contribution

It characterizes when these subgroup posets form chains, lattices, and their distributive or modular nature in finite cyclic and dihedral groups.

Findings

Poset is a chain only for p-groups.

Characterization of groups with a 2-element chain poset.

Poset forms a lattice in cyclic and dihedral groups.

Abstract

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this poset is C2, the chain with two elements. We also show that this poset forms a lattice in the case of finite cyclic and dihedral groups and give a characterization when this lattice is distributive and modular.