I'm doing this exercise in textbook Algebra by Saunders MacLane and Garrett Birkhoff
A commutator in a group $G$ is an element of the form $g^{-1} h^{-1}g h$. Then the group $[G, G]$ generated by the set of all commutators is normal in $G$.
I showed in this question that $N$ is normal in $G$ if and only if $\forall (n,g) \in N \times G: n^{-1} g^{-1} n g \in N$. Then it follows directly that $[G, G]$ is normal in $G$.
Now I try the the straightforward approach as follows. For $x = (g^{-1} h^{-1}g h)(a^{-1} b^{-1}a b) \in [G, G]$, our goal is to show $kxk^{-1} = kg^{-1} h^{-1}g ha^{-1} b^{-1}abk^{-1} \in [G, G]$. Then I'm stuck.
Could you please elaborate on how to proceed?
