Omega-1 is large

Statement

Let $G$ be a p-group, i.e., a group where the order of every element is a power of $p$. Then, the subgroup $\Omega _{1}(G)$ defined by:

$\Omega _{1}(G)=\langle x\in G\mid x^{p}=e\rangle$

Then, $\Omega _{1}(G)$ is a large subgroup of $G$: its intersection with every nontrivial subgroup of $G$ is nontrivial.