A Note on Lagrange Subsets of Finite Groups

TL;DR

This paper classifies finite groups where every Lagrange subset is a factor, showing such groups are limited to specific cyclic and elementary abelian groups, with a new proof approach.

Contribution

It provides a new, more direct proof of the classification of groups where all Lagrange subsets are factors, expanding understanding of group structure.

Findings

Nontrivial such groups are cyclic of prime order, order 4, or elementary abelian of order 4, 8, or 9.

The classification includes specific small cyclic and elementary abelian groups.

A new proof technique simplifies the existing classification proof.

Abstract

In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.