Omega subgroups are homomorph-containing

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