Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$? Let $G$ be a finite group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$.
Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?

Take any $g\in G$ then $|gA|=|A|>0.5|G|$ thus $(gA)\cap A\neq \emptyset .$ Let $c\in (gA)\cap A$ then $c=ga$ and therefore $g=ca^{-1}\in AA^{-1}.$
