Consider the free group $F_2 = ⟨x,y⟩$ and let $N$ be the smallest normal subgroup of $F_2$ containing $xyx^{−1}y^{−1}$.
$(a)$ Show that $x^2y^2x^{−2}y^{−2}$ is in $N$.
$(b)$ Show that any automorphism of $F_2$ sends $N$ to $N$.
For $(a)$ I think I have shown it and am wondering if it is valid. It goes as following:
Since N is a normal subgroup any two elements commute, and conjugation will be invariant. So, given $xyx^{-1}y^{-1}$ is an element of N, then:
$xxyx^{-1}y^{-1}x^{-1}$ is also in $N$ and $yxxyx^{-1}y^{-1}x^{-1}y^{-1}$ is also in $N$, and:
$yxxyx^{-1}y^{-1}x^{-1}y^{-1}=xyxyx^{-1}x^{-1}y^{-1}y^{-1}$ $=xxyyx^{-1}x^{-1}y^{-1}y^{-1}=x^2y^2x^{-2}y^{-2}$
Thus $x^2y^2x^{-2}y^{-2}$ is also in $N$
Is this a valid way to go about $(a)$?
As for $(b)$ I'm not really sure how to show this. Any help would be appreciated thank you.
Your reasoning is invalid since any two elements need not commute in $ N $, but a proof in a similar vein can be given.
Hint: $ xyx^{-1} y^{-1} = [x, y] $ is the commutator. What can you say about the quotient group $ F_2/N $?