  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 $$A^{-1}AK=KA^{-1}A=A^{-1}A\
\text{and}\ 6|K|\ge |A^{-1}A|=3|H|.$$
