Subset Product/Examples/Example 3
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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:
- $X X = G$
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $3$