Group theory: subset of a finite group Given


*

*$G$ be a finite group

*$X$ is a subset of group $G$

*$|X| > \frac{|G|}{2}$


I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to prove this？
If the third condition was $|X| = \frac{|G|}{2}$ instead, does the above statement still hold?
Thank you.

Note that for any $g \in G$ the set $g X^{-1}$ has also more than $\frac{|G|}{2}$ elements, hence $gX^{-1} \cap X \neq \emptyset$ so that there are $x,y \in X$ such that $gx^{-1} = y$, i.e. $g = yx$.
