Nilpotent Groups are Round

TL;DR

This paper introduces a new combinatorial property called roundness for finite groups and proves it is equivalent to the group being nilpotent, linking a geometric concept to an algebraic classification.

Contribution

It establishes a novel characterization of nilpotent groups via the concept of roundness, bridging combinatorial and algebraic properties.

Findings

Roundness is equivalent to nilpotence in finite groups.

The paper provides a new combinatorial perspective on group classification.

It offers a potential new tool for identifying nilpotent groups.

Abstract

We define a notion of roundness for finite groups. Roughly speaking, a group is round if one can order its elements in a cycle in such a way that some natural summation operators map this cycle into new cycles containing all the elements of the group. Our main result is that this combinatorial property is equivalent to nilpotence.