On some subgroup chains related to Kneser’s theorem

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 H ⊂ Stab(TS). Notice that Kneser’s Theorem only gives {0} 6= Stab(TS). 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.