Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$.
Please, I appreciate any help, since I have some ideas, but those are not clear.
2026-04-02 21:54:07.1775166847
Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$
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What you must prove is that for every $x \in G$ and $h \in H$, you have $xhx^{-1} \in H$. Now write $xhx^{-1} = xhx^{-1}h^{-1} \cdot h$. I used the "dot" before the last $h$ as a hint... what can you say about the RHS now? Does it bring to mind anything about $D(G)$, and the hypotheses that $D(G) \subset H$ and $H$ is a subgroup?