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Math Help - Sylow 2-subgroups

  1. #1
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    Sylow 2-subgroups

    I am trying to show that given a cyclic sylow 2-subgroup of a group of order 2^n is normal in G. I *think* this is true, but I am not sure and it makes my proof of a larger problem simpler if it were true. Anyone know if it is or isn't? I was trying to think of a quick example that would disprove it but couldn't. thanks in advance
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  2. #2
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    Quote Originally Posted by IthacaPride View Post
    I am trying to show that given a cyclic sylow 2-subgroup of a group of order 2^n is normal in G. I *think* this is true, but I am not sure and it makes my proof of a larger problem simpler if it were true. Anyone know if it is or isn't? I was trying to think of a quick example that would disprove it but couldn't. thanks in advance
    It is not true for T. That is the non-abelian group which is not A_4 or D_6.
    It has three cyclic Sylow 2-subgroups. Therefore they are not normal.

    EDIT: I am confused now, you said "... subgroup of a group of order 2^n". If the group has order 2^n then the Sylow 2-subgroup is the entire group itself. So it is definitely normal. But I do not think you really meant this.
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