Omega subgroups are homomorph-containing

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., omega subgroups of group of prime power order) always satisfies a particular subgroup property (i.e., homomorph-containing subgroup)}
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Statement

Suppose $G$ is a group of prime power order (i.e., a finite $p$-group for some prime number $p$). Then, the omega subgroups of $G$, defined as:

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

are homomorph-containing subgroups of $G$.

Related facts

• Omega subgroups not are subisomorph-containing
• Omega subgroups not are subhomomorph-containing