Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate: Difference between revisions

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(Created page with '{{subgroup property satisfaction| property = automorph-conjugate subgroup}} ==Statement== Let <math>F</math> be a free group on two generators, with <math>x,y</math> being ...')
 
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'''To prove''': <math>H</math> is automorph-conjugate in <math>F</math>.
'''To prove''': <math>H</math> is automorph-conjugate in <math>F</math>.


'''Proof''': By fact (2), the elementary Nielsen automorphisms of <math>F</math> generate <math>\operatorname{Aut}(F)</math>. We use a slightly modified generating set for <math>\operatorname{Aut}(F)</math> and show that this generating set
'''Proof''': By fact (2), the elementary Nielsen automorphisms of <math>F</math> generate <math>\operatorname{Aut}(F)</math>. We use this generating set to show that <math>H</math> is automorph-conjugate in <math>F</math> via fact (1):


* Replacing <math>x</math> by its inverse: <math>\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</math>.
* Replacing <math>x</math> by its inverse: <math>\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}Hx</math>.
* Replacing <math>y</math> by its inverse: <math>\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</math>.
* Swapping <math>x</math> and <math>y</math>: <math>\sigma([x,y]) = [y,x] = [x,y]^{-1}\in H</math>.
* <math>\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</math>.
* Replacing <math>x</math> by <math>xy</math>: <math>\eta([x,y]) = xyyy^{-1}x^{-1}y^{-1} = xyx^{-1}y^{-1} = [x,y] \in H</math>.
* <math>\sigma([x,y]) = [y,x] = [x,y]^{-1}\in \langle [x,y]\rangle</math>.

Revision as of 20:51, 28 April 2009

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Automorph-conjugate subgroup (?)) in a particular group or type of group .

Statement

Let F be a free group on two generators, with x,y being the generators. Let H be the subgroup of F generated by the commutator [x,y]=xyx−1y−1:

H=⟨[x,y]⟩.

Then, H is an automorph-conjugate subgroup of F.

Facts used

  1. Automorph-conjugate iff conjugate to image under a generating set of automorphism group
  2. Elementary Nielsen automorphisms generate the automorphism group of a finitely generated free group

Proof

Given: F is a free group with freely generating set {x,y}. H=⟨[x,y]⟩.

To prove: H is automorph-conjugate in F.

Proof: By fact (2), the elementary Nielsen automorphisms of F generate Aut(F). We use this generating set to show that H is automorph-conjugate in F via fact (1):

  • Replacing x by its inverse: τx([x,y])=x−1yxy−1=x−1yxy−1⋅x−1x=x−1[y,x]x=x−1[x,y]−1x∈x−1Hx.
  • Swapping x and y: σ([x,y])=[y,x]=[x,y]−1∈H.
  • Replacing x by xy: η([x,y])=xyyy−1x−1y−1=xyx−1y−1=[x,y]∈H.
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