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Math Help - Existence of Normal Subgroup

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    Existence of Normal Subgroup

    If G is a finite group and its p-Sylow subgroup P lies in the center of G, prove that there exists a normal subgroup N of G with P ^ N =e and PN=G
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    Quote Originally Posted by Chandru1 View Post

    If G is a finite group and its p-Sylow subgroup P lies in the center of G, prove that there exists a normal subgroup N of G with P \cap N =e and PN=G
    this is just a special case of Burnside's theorem usually called Burnside's normal complement theorem or Burnside's transfer theorem. the proof can be found almost anywhere:

    Burnside's theorem: if P is a Sylow subgroup of a group G and P \subseteq Z(N_G(P)), then there exists a normal subgroup N of G such that P \cap N=\{1\} and G=PN.
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