Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that $\displaystyle
\left| G \right| \geqslant \left| S \right| + \left| T \right|
$
There are |S| elements in S. There are |T| elements in $\displaystyle T^{-1}$, and also in its coset $\displaystyle gT^{-1}$. Also, the sets S and $\displaystyle gT^{-1}$ are disjoint: for suppose that an element $\displaystyle s\in S$ is also in $\displaystyle gT^{-1}$. Then $\displaystyle s = gt^{-1}$ for some $\displaystyle t\in T$. But that implies that $\displaystyle st=g$, which contradicts the choice of g. Therefore there are |S|+|T| distinct elements in the set $\displaystyle S\cup gT^{-1}$, and so $\displaystyle |G|\geqslant |S|+|T|$.