# Subset Product/Examples/Example 3

## Example of Subset Product

Let $G$ be a group.

Let the order of $G$ be $n \in \Z_{>0}$.

Let $X \subseteq G$ be a subset of $G$.

Let $\card X > \dfrac n 2$.

Then:

$X X = G$

where $X X$ denotes subset product.

## Proof

Let $g \in G$ be arbitrary.

Let $Y := \set {x^{-1} g: x \in X}$.

Then:

$\card Y = \card X$

As $\card X > \dfrac {\order g} 2$ we have:

$X \cap Y \ne \O$

So:

$\exists x_1, x_2 \in X: x_1^{-1} g = x_2$

That is:

$g = x_1 x_2$

and so:

$g \in X X$

As $g$ is arbitrary, it follows that:

$X X = G$

$\blacksquare$