Question
Let $H$ be a normal subgroup of a finite group $G$ and let $x \in G$. If $\operatorname{gcd}(|x|,|G / H|)=1$, show that $x \in H$. (This exercise is referred to in Chapter 25.)
Step 1
Since $H$ is a normal subgroup of $G$, we can form the quotient group $G/H$. Show more…
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Let $G$ be an abelian group. Let $n$ be a fixed integer, and let $H=\left\{x \in G: x^{n}=e\right\}$. Prove that $H$ is a subgroup of $G$.
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