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.

Answer for the additional question: the statement need not be true if $|X|=|G|/2$.
For example, let $G$ be the cyclic group of order $2$ and $X$ consist of the non-identity element.
A (slightly) more general example: let $G$ be the cyclic group of order $2n$, let $g$ be a generator, and let
$$X=\{g^{2k}\mid k=0,1,\ldots,n-1\}\ .$$
And a bit more general again: if $X$ is a proper subgroup (not just a subset) of $G$ then the product of elements of $X$ is still in $X$, and hence not all elements of $G$ will be obtained.
