Groups of order 21 (Need help understanding an inference)

[Samuelb88]

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 x (of order 7) and y (of order 3) generate the entire group G. Since H is normal, we have know that yxy^{-1} = x^k, for some k, 0 \leq k \leq 6. k cannot equal 0 and 1 because that would imply x = e 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 y has order 3, and y^3 x y^{-3} = x^{k^3} ...

How did he infer that y^3 x y^{-3} = x^{k^3} from what was given?

 

[micromass]

Well, first notice that for every n, it holds that

(yxy^{-1})^n=yx^ny^{-1}

thus

<br /> \begin{eqnarray*}<br /> x^{k^3}<br /> &amp; = &amp; ((x^k)^k)^k\\<br /> &amp; = &amp; ((yxy^{-1})^k)^k\\<br /> &amp; = &amp; y(x^k)^ky^{-1}\\<br /> &amp; = &amp; y(yxy^{-1})^k y^{-1}\\<br /> &amp; = &amp; y^2x^ky^{-2}\\<br /> &amp; = &amp; y^3 x y^{-3}<br /> \end{eqnarray*}<br />

Does that help??

 

[Samuelb88]

Oh doh! I forgot about that property. So to answer your question, yes it does help! Thank you very much!