Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Groups of order 21 (Need help understanding an inference)

  1. Aug 18, 2011 #1
    Sylow's theorem tells us that there is one 7-Sylow subgroup and either one of seven 3-Sylow subgroups. Call these subgroups H and K respectively. Sylow's theorem also tells that H is normal in G.

    I'm not going to write it all out as I don't think it's necessary but in the case when we have seven 3-Sylow subgroups, we conclude that the generators [itex]x[/itex] (of order 7) and [itex]y[/itex] (of order 3) generate the entire group G. Since H is normal, we have know that [itex]yxy^{-1} = x^k[/itex], for some [itex]k[/itex], [itex]0 \leq k \leq 6[/itex]. [itex]k[/itex] cannot equal 0 and 1 because that would imply [itex]x = e[/itex] in the first case and G is abelian in the second case, contrary to assumption in both cases.

    Here's where I get lost:

    Since [itex]y[/itex] has order 3, and [itex]y^3 x y^{-3} = x^{k^3}[/itex] ...

    How did he infer that [itex]y^3 x y^{-3} = x^{k^3}[/itex] from what was given?
  2. jcsd
  3. Aug 18, 2011 #2
    Well, first notice that for every n, it holds that



    & = & ((x^k)^k)^k\\
    & = & ((yxy^{-1})^k)^k\\
    & = & y(x^k)^ky^{-1}\\
    & = & y(yxy^{-1})^k y^{-1}\\
    & = & y^2x^ky^{-2}\\
    & = & y^3 x y^{-3}

    Does that help??
  4. Aug 18, 2011 #3
    Oh doh! I forgot about that property. So to answer your question, yes it does help! Thank you very much!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook