If G is a finite goup and let $\displaystyle A \subseteq G$. if $\displaystyle |A|> |G|/2$ then prove that for every $\displaystyle g \in G$ there exists [tex]h,k \in A [/Math] such that $\displaystyle hk=g$ (Headbang)

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- Feb 24th 2009, 01:29 AMChandru1Subset of order > 1/2 of order G
If G is a finite goup and let $\displaystyle A \subseteq G$. if $\displaystyle |A|> |G|/2$ then prove that for every $\displaystyle g \in G$ there exists [tex]h,k \in A [/Math] such that $\displaystyle hk=g$ (Headbang)

- Feb 24th 2009, 09:13 AMThePerfectHacker
For any $\displaystyle x$ there exists a unique $\displaystyle y$ so that $\displaystyle xy=g$, so we will define $\displaystyle \widehat x=y$. Assume that for any $\displaystyle a\in A$ we have $\displaystyle \widehat a\not \in A$ i.e. for any $\displaystyle a\in A \implies \widehat a \in G-A$. Since $\displaystyle |A| > |G-A|$ by pigeonhole principle there exists two elements, $\displaystyle a,b\in A, a\not = b$ that satisfy $\displaystyle \widehat a = \widehat b$ but this means $\displaystyle a=b$, a contradiction.