1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted