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:
 * $X \cap Y \ne \0$

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:
 * $X X = G$

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