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We know that if $H\leq G$, then $H^{G}=\left\langle h^{g};h\in H\text{ and }g\in G\right\rangle \trianglelefteq G,$ is the normal closure of $H$ in $G.$

Usually, when we kill $T\trianglelefteq G$ in $G/T$ , we have some property in $G/T.$

Example: If $T$ contains all the commutators of $G,$ then $G/T$ is abelian.

My question is: what can we say about the group $G/H^{G}$? What important properties does it have?

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    (So, for example, all abelian groups satisfy the law $[x, y]=1$ while all groups of exponent $e$ satisfy the law $x^e$.)2012-08-29

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Let's generalize from your abelianization example. Commutators are elements of the form $aba^{-1}b^{-1}$, and a group being abelian means any element of the form $aba^{-1}b^{-1}$ is trivial. But there's nothing special about that form. You can write any form you like, such as $abc^2ba^{-1}$. If $H$ contains all elements of that form, then all such elements will be trivial in the quotient group.

This is closely related to the concept of a group presentation.

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    Note that $H^G=H[G,H]$2012-08-29
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There can be no special properties, since every normal subgroup $N$ arises in this way as $N^G$.

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One fact is the following. If $G$ is finite then $G$ must have a nilpotent subgroup $H$ with $H^G=G$. This can be seen by induction on $|G|$. If $G$ has a maximal subgroup $H$ that is not normal in $G$, then the result follows by induction applied to $H$. Otherwise all maximal subgroups are normal, which implies that $G$ itself is nilpotent.

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    Lima: thank you very much!2012-08-30