Intersection of DPICF and direct factor is central factor

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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Statement

Verbal statement

The intersection of a direct projection-invariant central factor with a direct factor is a central factor.

Proof

Hands-on proof

Suppose $H$ is a direct projection-invariant central factor of $G$, and $K$ is a direct factor of $G$. We need to show that $H\cap K$ is a central factor of $G$.

Since the property of being a central factor is transitive, and $K$ is already a central factor, it sufices to show that $H\cap K$ is a central factor of $K$.

In order to do this, consider a direct projection $\pi :G\to K$ corresponding to the direct factor $K$. Since $H$ is direct projection-invariant, it must map to inside $H\cap K$. Further, the elements of $H\cap K$ are $\pi$-invariant, and hence the image of $H$ under $\pi$ is in fact $H\cap K$.

Now, since $H$ is a central factor of $G$, $\pi (H)$ is a central factor of $\pi (G)$, telling us that $H\cap K$ is a central factor of $K$ and we are done.