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Math Help - finite group

  1. #1
    mms
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    finite group

    Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that <br />
\left| G \right| \geqslant \left| S \right| + \left| T \right|<br /> <br />
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  2. #2
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    Quote Originally Posted by mms View Post
    Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that <br />
\left| G \right| \geqslant \left| S \right| + \left| T \right|<br /> <br />
    Take an element g\notin ST and consider the set S\cup gT^{-1}.
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  3. #3
    mms
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    Could you help me a little bit more? i still can't do this problem >.<

    thanks!
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  4. #4
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    There are |S| elements in S. There are |T| elements in T^{-1}, and also in its coset gT^{-1}. Also, the sets S and gT^{-1} are disjoint: for suppose that an element s\in S is also in gT^{-1}. Then s = gt^{-1} for some t\in T. But that implies that st=g, which contradicts the choice of g. Therefore there are |S|+|T| distinct elements in the set S\cup gT^{-1}, and so |G|\geqslant |S|+|T|.
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  5. #5
    mms
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    ahh i see, thank you!
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