Perfect core is 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., perfect core) always satisfies a particular subgroup property (i.e., homomorph-containing subgroup)}
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Statement

Suppose $G$ is a group and $H$ is the perfect core of $G$: it is the unique largest perfect subgroup of $G$. Then, $H$ is a homomorph-containing subgroup of $G$: given any homomorphism $\varphi :H\to G$, we have $\varphi (H)\leq H$.

Related facts

• Fitting subgroup is normal-isomorph-free in finite
• Solvable core is normal-isomorph-free in finite

Facts used

1. Perfectness is quotient-closed: The image of a perfect group under a surjective homomorphism is perfect.

Proof

Given: A group $G$ with perfect core $H$ and a homomorphism $\varphi :H\to G$.

To prove: $\varphi (H)\leq H$.

Proof: $\varphi (H)$ is perfect by fact (1). By the definition of perfect core, any perfect subgroup of $G$ is contained in $H$. Thus, $\varphi (H)\leq H$.