Order of Finite p-Group is Power of p/Proof 2

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Theorem

Let $G$ be a finite group.

Let $p$ be a prime number.

Let all elements of $G$ have order a power of $p$.


Then $G$ is a $p$-group.


Proof

Let every element of $G$ be a $p$-element.

Let $q$ be a prime number which is a divisor of the order $\order G$ of $G$.

By Cauchy's Lemma (Group Theory), there exists an element of $G$ whose order is a divisor of $q$.

But as the order of all elements of $G$ divide $p^n$ it follows that $q = p$.

Thus $G$ is a group whose order is $p^n$ for some $n \in \Z_{>0}$.

$\blacksquare$