Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate

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 $[x,y]=xy{x}^{-1}{y}^{-1}$:

$H=⟨[x,y]⟩$.

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 $\mathrm{A}\mathrm{u}\mathrm{t}(F)$. We use a slightly modified generating set for $\mathrm{A}\mathrm{u}\mathrm{t}(F)$ and show that this generating set

• Replacing $x$ by its inverse: ${\tau }_x([x,y])=x^{-1}yx{y}^{-1}=x^{-1}yx{y}^{-1}\cdot x^{-1}x=x^{-1}[y,x]x=x^{-1}[x,y{]}^{-1}x\in x^{-1}⟨[x,y]⟩x$.
• Replacing $y$ by its inverse: ${\tau }_y([x,y])=x{y}^{-1}{x}^{-1}y=y^{-1}y\cdot x{y}^{-1}{x}^{-1}y=y^{-1}[y,x]y=y^{-1}[x,y{]}^{-1}y\in y^{-1}⟨[x,y]⟩y$.
• $\eta ([x,y])=y^{-1}x{y}^{-1}{x}^{-1}yy=y^{-1}\cdot {\tau }_y([x,y])\cdot y=y^{-2}[x,y]y^2\in y^{-2}⟨[x,y]⟩y^2$.
• $\sigma ([x,y])=[y,x]=[x,y{]}^{-1}\in ⟨[x,y]⟩$.