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.