TITLE: If $X$ is the set of all group elements of order $p$, and $X$ is finite, then $\langle X \rangle$ is finite
QUESTION [5 upvotes]: Suppose $G$ is any group and $X$ is the set of all elements of order $p$ in that group where $p$ is a prime. Prove that if $X$ is finite, then $\langle X\rangle$ is finite, where $\langle X\rangle$ is group generated by the elements of $X$.

If possible, I'm interested in a solution that assumes as few theorems as possible (aside from the well known ones of course).

REPLY [3 votes]: Another way to prove this is by applying a classical result called Dietzmann's Lemma (for a proof see for example the book of Derek Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1.) It implies that if you have a finite set $X$ in a group $G$, only containing elements of finite order and is closed under conjugation, then $\langle X \rangle$ is finite.