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If X is a subset (not a subgroup) of a finite group G, and X contains more than half of the elements in G. Show that for each element $g \in G \ ; g=ab \ ; a,b \in X$
My intuition is if you treat X like a coset, $xX$ it will generate G as you vary x. I'm not sure how to show that though. A hint given was that no theory is required.

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marked as duplicate by mesel, Dietrich Burde, Community Nov 19 '15 at 20:44

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Let $X^{-1}$ be the set of inverses of elements of $X$.

For any $g\in G$, $(gX^{-1})\cap X$ is non-empty (Pigeonhole Principle). So $gb^{-1}=a$ for some $a$, $b\in X$. The result follows.

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