Two Inverse results

TL;DR

This paper investigates the structure of subsets within groups where the product set is small, revealing the existence of specific subgroups and elements that characterize the subset's algebraic properties.

Contribution

It provides inverse results characterizing the structure of subsets with small product sets in groups, extending classical sumset theory to non-abelian groups.

Findings

Existence of an element and a proper subgroup with specific product set properties.

Structural description involving normal subgroups and subgroup inclusions.

Bounds relating subset size and subgroup sizes.

Abstract

Let $ A$ be a subset of group $G_0$ with $|{A^{-1}A}|\le 2|A|-2.$ We show that there are an element $a\in A$ and a non-null proper subgroup $H$ of $G$ such that one of the following holds: \begin{itemize} \item $x^{-1}Hy \subset A^{-1}A,$ for all $(x,y)\in A^2\setminus (Ha)^2,$ \item $xHy^{-1} \subset AA^{-1},$ for all $(x,y)\in A^2\setminus (aH)^2.$ \end{itemize} where $G$ is the subgroup generated by ${A^{-1}A}.$ Assuming that $A^{-1}A\neq G$ and that $ |A^{-1}A|< \frac{5|A|}3,$ we show that there are a normal subgroup $K$ of $G$ and a subgroup $H$ with $K\subset H\subset A^{-1}A $ and $2|K|\ge |H|$ such that $$A^{-1}AK=KA^{-1}A=A^{-1}A\ \text{and}\ 6|K|\ge |A^{-1}A|=3|H|.$$