Subset of order > 1/2 of order G

**Chandru1** · February 24th 2009, 01:29 AM

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

**ThePerfectHacker** · February 24th 2009, 09:13 AM

Originally Posted by Chandru1

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

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

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