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Let $G$ be a group and $H$ a subgroup of $G$. Suppose $g^2\in H$ for all $g\in G$. Show $H$ is a normal subgroup of $G$. I tried lots of methods, but failed. Any suggestion? Thanks.

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marked as duplicate by Jared, Micah, Dan Rust, Potato, Hanul Jeon Jul 25 '13 at 23:42

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up vote 10 down vote accepted

Let $g \in G$, $h \in H$. We know that $g^2 \in H$. Hence $g^2 h \in H$. We have: $$ (ghg^{-1})(g^2 h) = ghgh = (gh)^2 \in H $$

Thus, $ghg^{-1} \in H$ and $H$ is a normal subgroup of $G$.

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