# Hyperballeans of groups

## TL;DR

This paper introduces ballean structures on the power set of groups, especially focusing on the lattice of subgroups, to explore the relationship between algebraic and geometric properties of groups.

## Contribution

It defines new ballean structures on subgroup lattices and investigates how these structures reflect the algebraic relationships between different groups.

## Key findings

- Ballean structures can encode subgroup lattice properties.
- Relations between groups can be studied via their subgroup balleans.
- New connections between algebraic and geometric group properties are established.

## Abstract

In this paper we define some ballean structure on the power set of a group and, in particular, we study the subballean with support the lattice of all its subgroups. If $G$ is a group, we denote by $L(G)$ the family of all subgroups of $G$. For two groups $G$ and $H$, we relate their algebraic structure via the ballean structure of $L(G)$ and $L(H)$.