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Math Help - Subset of order > 1/2 of order G

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    Subset of order > 1/2 of order G

    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
    Last edited by ThePerfectHacker; February 24th 2009 at 09:53 AM.
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    Quote Originally Posted by Chandru1 View Post
    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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