Question
Let $K$ be a Sylow $p$ -subgroup of a finite group $G$. Prove that if $x \in$ $N(K)$ and the order of $x$ is a power of $p$, then $x \in K .$ (This exercise is referred to in this chapter.)
Step 1
A Sylow $p$-subgroup $K$ of a finite group $G$ is a maximal $p$-subgroup of $G$, meaning that there is no larger subgroup of $G$ whose order is a power of $p$. Show more…
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Let $G$ be an abelian group. Let $H$ be a subgroup of $G$, and let $K$ consist of all the elements $x$ in $G$ such that some power of $x$ is in $H$. That is, $K=\left\{x \in G:\right.$ for some integer $\left.n>0, x^{n} \in H\right\}$. Prove that $K$ is a subgroup of $G$.
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