I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$.
I've tried to argue by contradiction: suppose it had an element $g$ of order $p^2$, then the subgroup generated by $g$ has index $p$ so it is normal, and from there I would like to find an element $h$ of order $p$ whose cyclic subgroup has trivial intersection with $ \langle g \rangle$. Then $G$ is the semi-direct product of $\langle g \rangle$ and $ \langle h \rangle$, and I am hoping this will give me a contradiction by telling me that $G$ is abelian or something similar.
Any help is appreciated!