# On Some Subgroup Chains Related to Kneser's Theorem

## TL;DR

This paper explores subgroup chains related to Kneser's Theorem, extending results from abelian to nonabelian groups using additive number theory and poset properties.

## Contribution

It introduces an analogous poset for nonabelian groups and extends key results of Balandraud to this broader context.

## Key findings

- Short proof of Balandraud's results in abelian groups
- Extension of subgroup chain properties to nonabelian groups
- Use of classical additive number theory tools

## Abstract

A recent result of Balandraud shows that for every subset S of an abelian group G, there exists a non trivial subgroup H such that |TS| <= |T|+|S|-2 holds only if the stabilizer of TS contains H. Notice that Kneser's Theorem says only that the stabilizer of TS must be a non-zero subgroup. This strong form of Kneser's theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud's results in the abelian case.