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Sylow Theorem

  1. Aug 25, 2010 #1
    1. The problem statement, all variables and given/known data

    Let G be a group of order 324. Show that G has subgroups of order 2,
    3, 4, 9, 27 and 81, but no subgroups of order 10.

    2. Relevant equations

    Sylow showed that if a prime power [itex]p^k [/itex] divides the order of a finite group G, then G has a subgroup of order [itex]p^k [/itex].

    3. The attempt at a solution

    I can see that G can have the subgroup 2 because [itex]2^n n=1 = 2[\latex]
    subgroup 3 because [itex]3^n n=1 = 3[\latex] divides 324
    subgroup 4 because [itex]2^n n=2 = 4[\latex] divides 324
    subgroup 9 because [itex]3^n n=2 = 9[\latex] divides 324
    subgroup 27 because [itex]3^n n=3 = 27[\latex] divides 324
    subgroup 81 because [itex]3^n n=4 = 81[\latex] divides 324

    I know that 10 does not divide 324 in Z

    Is that enough to show that the sub group can't be order 10 ?
  2. jcsd
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