# If $|H| \leq \frac{1}{1- \epsilon}|A|$ then H is a subgroup

I'm studying a topic in group theory, I'm stuck with the following point.

$G$ is a finite group, $A \subset G$ a subset. We have $H= AA^{-1} = A^{-1}A$, and we know that $$|H| \leq \frac{1}{1- \varepsilon}|A| \ ,\ \ \ \ \varepsilon > 0.$$

Then if $\varepsilon < \frac{1}{2}$, given $x , y \in H$ there are representations $x = dc^{-1}$ and $y = ef^{-1}$ with $c = e$.

Why this is true ?

Notice that $|H|\lt 2|A|$, hence $$|xA\cup yA|\gt 2|H|-|xA\cap yA|.$$ Since $|H|\geqslant |A|$ and $|xA\cup yA|\leqslant 2|A|$ we have $xA\cap yA\neq \emptyset$.
• sorry but i don't understand why your inequality implies $xA \cap yA \neq \varnothing$ – WLOG Feb 23 '14 at 22:08