The posets of classes of isomorphic subgroups of finite groups

TL;DR

This paper introduces and investigates the poset of isomorphism classes of subgroups of finite groups, revealing how this structure can determine the group's overall structure in specific cases.

Contribution

It defines a new poset based on subgroup isomorphism classes and explores its properties and implications for understanding finite groups.

Findings

Poset contains the lattice of solitary subgroups.

In certain cases, the poset determines the structure of the group.

Provides new insights into subgroup classification and group structure.

Abstract

In this paper we introduce and study the poset of equivalence classes of subgroups of a finite group $G$, induced by the isomorphism relation. This contains the well-known lattice of solitary subgroups of $G$. We prove that in several particular cases it determines the structure of $G$.