Talk:Automorph-conjugate subgroup

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Although the term intravariant subgroup has been used for this subgroup property at a few places in the literature, we're sticking to automorph-conjugate because there has been no paper in the last 30 years using the old term, and I believe that automorph-conjugate is more self-explanatory. Searches on the old term will still yield results in the wiki. Vipul 18:02, 22 May 2008 (UTC)

An example

I’m not sure if examples are meant to be included in acticles about terms, and if so, whether this example is interesting enough. If it is, please copy it to the article:

The group \langle [x,y]\rangle is an automorph-conjugate subgroup of the Free Group F2 generated by x and y. This can be shown by checking that image of [x,y] under each of a set of generators of Aut(F2) is a conjugate to [x,y]. Here, the generators τx,τy,σ,η, due to Armstrong, Forrest, Vogtmann [1], are used:

  • \tau_x([x,y]) = x^{-1}yxy^{-1} = x^{-1}yxy^{-1} \cdot x^{-1}x = x^{-1}[y,x]x = x^{-1}[x,y]^{-1}x \in x^{-1}\langle [x,y]\rangle x
  • \tau_y([x,y]) = xy^{-1}x^{-1}y = y^{-1}y\cdot xy^{-1}x^{-1}y = y^{-1}[y,x]y = y^{-1}[x,y]^{-1}y \in y^{-1}\langle [x,y]\rangle y
  • \sigma([x,y]) = [y,x] = [x,y]^{-1}\in \langle [x,y]\rangle
  • \eta([x,y]) = y^{-1}xy^{-1}x^{-1}yy = y^{-1}\cdot \tau_y([x,y]) \cdot y = y^{-2} [x,y] y^2 \in y^{-2}\langle [x,y]\rangle y^2
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