ON KNESER'S ADDITION THEOREM IN GROUPS

The following theorem is proved. Theorem A. Let G be a group written additively with finite non- empty subsets A, B. Assume that B is commutative, i.e. b1 + b.1 = bi+bi, for ?>1% b.,e B. Then there exists an Abelian subgroup H of G such that A+B+H=A+H+B=A+B and \A+B\>\A + H\ + \B+H\-\H\. This is Kneser's theorem, if G is Abelian. Also, as an application of the above theorem, the following is proved. Theorem B. Let G be a finite group of order v (v>l) and let ai, ■ ■ ■ , ar be a sequence (repeats are allowed) of nonzero elements oj'G. The set S of sums a?1 + - ■ '+#,, where la&<- ' "0<=» oid l^t^v must contain a nontrivial subgroup H of G. Finally, the Kemperman (/-transform, a transform similar to the Dyson e-transform, is introduced and evidence is given to support the conjecture that Theorem A remains true, if the commutativity of B is dropped.