Recovering information about a finite group from its subrack lattice

TL;DR

This paper demonstrates that the subrack lattice of a finite group encodes key structural properties, such as nilpotence class and p-nilpotence, by analyzing the lattice's atoms and their corresponding element orders.

Contribution

It establishes that the subrack lattice uniquely determines the nilpotence class and p-nilpotence of finite groups under certain conditions, revealing new insights into group structure from lattice data.

Findings

Subrack lattice determines the nilpotence class of a finite group.

The lattice can be used to estimate the orders of elements at atoms.

The subrack lattice can identify p-nilpotence under specific conditions.

Abstract

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines $p$-nilpotence of the group if a certain condition is met.