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Let $G$ be a group and let $G'=\langle aba^{-1}b^{-1}\mid a,b\in G\rangle$. In other words, it is the subgroup of all finite products of elements in $G$ of the form $aba^{-1}b^{-1}$. How to prove that $G'$ is a normal subgroup of $G$?

So if $g$ is an arbitrary element of $G$, I want to show that $g(a_1b_1a_1^{-1}b_1^{-1})(a_2b_2a_2^{-1}b_2^{-1})...(a_nb_na_n^{-1}b_n^{-1})g^{-1}$ can be written as the finite product of elements of the form $aba^{-1}b^{-1}$. How to approach this?

Arturo Magidin
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    Hint: Start by verifying that $g[a,b]g^{-1}=[gag^{-1}, gbg^{-1}]$. – Mark Nov 07 '24 at 19:08
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    This is a good example of a problem for which it is easier to prove a stronger result: the commutator subgroup of $G$ is a characteristic subgroup of $G$. – Derek Holt Nov 07 '24 at 20:28

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