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 $n \in \Z_{>0}$.
$\blacksquare$