1. Not finding help here? Sign up for a free 30min 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!

Abstract Algebra: Groups of order 21

  1. Apr 5, 2005 #1
    I was given a problem to prove there are at most 3 groups of order 21, with extra credit for proving there are at most 2. I am pretty stuck on this one but here is what I have so far:

    Suppose G is a group of order 21
    Let K be a sylow 3-subgroup of G and let H be a sylow 7-subgroup of G.

    By Sylow's third theorem, H is a normal subgroup because it can only be expressed one way such that x (mod p) = 1 where x divides the order of G (x = 1 only). By Sylow's Third Theorem, G has either 1 or 7 sylow 3-subgroups.

    2 cases now-

    For K being a unique sylow 3-subgroup (G has only 1 3-subgroup), H and K are both normal in G. Being prime ordered, they are also both cylic, so let H = <x> and K <y>, then
    [tex]xyx^{-1}y^{-1} = (xyx^{-1})y^{-1} \in Ky^{-1} = K[/tex] but also
    [tex]xyx^{-1}y^{-1} = x(yx^{-1}y^{-1} \in xH = H[/tex] so
    [tex]xyx^{-1}y^{-1} \in K \cap H = \{e\}[/tex]
    which essentially means for [tex]xy = yx \in HK[/tex]

    The last conclusion coming from the fact that H and K are relatively prime so no elements besides the identity overlap. That being established, HK is an abelian group

    -I have 2 problems now. First of all, I don't know how to tie HK back to G, and second of all I'm not sure how to proceed if K is not a normal subgroup (ie 7 subgroups of order 3 exist). Any advice would be much appreciated!

    Edit: I proved that G cannot have 7 subgroups of order 3, but the first question still remains. Thanks!
     
    Last edited by a moderator: Apr 5, 2005
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Abstract Algebra: Groups of order 21
Loading...