On Kneser’s addition theorem in groups

Abstract:

The following theorem is proved. Theorem A. Let $G$ be a group written additively with finite nonempty subsets $A,B$. Assume that $B$ is commutative, i.e. ${b_1} + {b_2} = {b_2} + {b_1}$, for ${b_1},{b_2} \in B$. Then there exists an Abelian subgroup $H$ of $G$ such that $A + B + H = A + H + B = A + B$ and $|A + B| \geqq |A + H| + |B + H| - |H|$. 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 $\upsilon (\upsilon > 1)$ and let ${a_1}, \cdots ,{a_\upsilon }$ be a sequence (repeats are allowed) of nonzero elements of $G$. The set $S$ of sums ${a_{{i_1}}} + \cdots + {a_{{i_t}}}$ where $1 \leqq {i_1} < \cdots < {i_t} \leqq \upsilon$ and $1 \leqq t \leqq \upsilon$ must contain a nontrivial subgroup $H$ of $G$. Finally, the Kemperman $d$-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.