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P-Sylow subgroup congruence

  1. Dec 8, 2008 #1
    Let [tex]p[/tex] be a prime, [tex]G[/tex] a finite group, and [tex]P[/tex] a [tex]p[/tex]-Sylow subgroup of [tex]G[/tex]. Let [tex]M[/tex] be any subgroup of [tex]G[/tex] which contains [tex]N_G(P)[/tex]. Prove that [tex][G:M]\equiv 1[/tex] (mod [tex]p[/tex]). (Hint: look carefully at Sylow's Theorems.)
     
  2. jcsd
  3. Dec 8, 2008 #2

    mathwonk

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    since the word N = normalizer occurs, one is led to look at the action on G on p by conjugation. G permutes subgroups of G and we look at the orbit of P. this orbit contains kp+1 subgroups, so the theorem holds if N = M. then what?
     
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