Let $G$ be a finite group and $S$ a nonempty subset of $G$. I want to prove (or disprove) that $S^{|G|}$ (that is products of length $|G|$ of elements of $S$) is a subgroup.
My work so far :
Since we are in a finite group, it suffices to show that $S^{|G|}$ is stable under the group product.
When $e\in S$, we have $S^{|G|} = \bigcup_{k=0}^{|G|}S^{k}$ and this is the subgroup generated by $S$.
However I'm having issues solving this when $e\notin S$.
I've tried to consider the monoid $C\subset \Bbb N$ of length of products of elements equal to the neutral element. If $S^{|G|}$ is not a group, then there exists $k\notin C$ smaller than $|G|$ such that $k+|G|\in C$. However I don't know where to go from here.
Any help is appreciated !
Edit : Note that I want products of exactly $|G|$ elements of $S$, not products of at most $|G|$ elements of $S$