Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate
From Groupprops
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 = \langle [x,y] \rangle](/w/images/math/1/c/5/1c5327191f29a1c558ec3ed489cfa288.png)
.
Then, H is an automorph-conjugate subgroup of F.
Facts used
- Automorph-conjugate iff conjugate to image under a generating set of automorphism group
- 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}. .
To prove: H is automorph-conjugate in F.
Proof: By fact (2), the elementary Nielsen automorphisms of F generate 
. We use a modified version of this generating set to show that H is automorph-conjugate in F via fact (1):
- Replacing x by its inverse:
![\tau_x([x,y]) = [x^{-1},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](/w/images/math/7/7/3/7738a4fc01bc589a2a52ebb074814088.png)
.
- Replacing y by its inverse: τy([x,y]) = [x,y − 1] = xy − 1x − 1y = y − 1[x,y] − 1yiny − 1Hy.
- Swapping x and y:
![\sigma([x,y]) = [y,x] = [x,y]^{-1}\in H](/w/images/math/2/f/8/2f897cb5c28f4ce38fe61e427c90e7fe.png)
.
- Replacing x by xy:
![\eta([x,y]) = xyyy^{-1}x^{-1}y^{-1} = xyx^{-1}y^{-1} = [x,y] \in H](/w/images/math/1/b/2/1b27dd7ebf6cb501142ba20a33ecca29.png)
.
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