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First, I was asked to show that if $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and $G/G'$ is Abelian.

This was not too difficult to show.

The second part of the question said if $G$ is a group and $H\supseteq G'$, where $G'$ is as in the last part, then $H\trianglelefteq G$ and $G/H$ is Abelian.

I'm not sure of the best way to approach this problem. Any help would be appreciated.

Michael Shumate
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1 Answers1

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Hint for normality: for all $g \in G$ and $h \in H$, $g^{-1}hg=[g,h^{-1}]h$.

Nicky Hekster
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