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Math Help - quotient group

  1. #1
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    quotient group

    I know that a group H has exactly one 3-Sylow subgroup L. Also, |H|=30. And |M|=15 is normal.

    Now I should consider the quotient group H/M and prove that \forall a \in H a^2 \in M. And then that each element of odd order in H is in M.

    Can someone help me please?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    I don't know - it depends on what you are trying to prove...
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  3. #3
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    Quote Originally Posted by HoneyPi View Post
    I know that a group H has exactly one 3-Sylow subgroup L. Also, |H|=30. And |M|=15 is normal.

    Now I should consider the quotient group H/M and prove that \forall a \in H a^2 \in M. And then that each element of odd order in H is in M.

    Can someone help me please?

    For (1): Lemma: if N\triangleleft G\,\,\,and\,\,\,[G:N]=n , then g^n\in N\,,\,\forall\,g\in G ( for the proof use Lagrange in the quotient G/H )

    For (2): Use directly Lagrange and (1).

    Tonio
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