I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?
Group Theory – Element of Order p in Groups of Prime Power Order
finite-groupsgroup-theoryp-groups
Best Answer
This follows immediately from Lagrange theorem, you don't need any stronger result.
If the order of the group is $p^k$ with $k \neq 0$, then by Lagrange Theorem, the order of any element divides $p^k$.
Pick some $x \in G, x \neq e$. Then the order of $x$ is $p^m$ with $1 \leq m \leq k$. Let
$$y:=x^{p^{m-1}} \,.$$
Prove that the order of $y$ is $p$.