Omega-1 is large

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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.