Results 1 to 4 of 4

Math Help - Sylow group question

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    2

    Sylow group question

    1.Let G be a group with |G|=p^2*q,where p and q are distinct primes,show that G has a normal Sylow p-subgroups or a normal Sylow q-subgroups.

    2.if |G| < 100 and G is non-abelian and simple,then show that |G|=60.

    Please help to solve these two questions..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by seng View Post
    1.Let G be a group with |G|=p^2*q,where p and q are distinct primes,show that G has a normal Sylow p-subgroups or a normal Sylow q-subgroups.
    let mp+1 and nq+1 be the number of Sylow p-subgroups and q-subgroups respectively. we just need to show that either

    m = 0 or n = 0. so suppose, on the contrary, that m > 0 and n > 0 and consider two possible cases:

    Case 1: mp+1=q, \ nq+1=p. then we'll have q>p and p>q, which is impossible.

    Case 2: mp+1=q, \ nq+1=p^2. thus, since every two Sylow q-subgroups intersect in the identity element only, we'll have:

    p^2q=|G| > p^2(q-1) + p^2 = p^2q, which is impossible. Q.E.D.

    2.if |G| < 100 and G is non-abelian and simple,then show that |G|=60.
    to show this, you'll need more than just this fact that for primes p, q, groups of order p^n, \ n >1, or pq or p^2q are not simple!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    2
    Actually i not really undestand how to do the second question,can you guide me to do?
    I also will try my best to do...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by seng View Post
    Actually i not really undestand how to do the second question,can you guide me to do?
    I also will try my best to do...
    Just write out the numbers: 61,62,...,100.
    And start out canceling the ones you know are not it.
    Cross out the primes because you are looking for the non-abelian ones.
    Now cross out the prime powers because p-groups.
    Now cross out the pq ones.
    And then the p^2q ones.
    And see if you can get rid of all those numbers.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. p-sylow group abelian
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 22nd 2011, 08:24 PM
  2. Sylow p-group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 6th 2009, 08:03 PM
  3. Group Theory - Sylow and Conjugation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 21st 2009, 10:15 AM
  4. Sylow group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 6th 2008, 12:08 PM
  5. Sylow p-group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 9th 2008, 09:33 AM

Search Tags


/mathhelpforum @mathhelpforum