Central factor satisfies image condition

Statement with symbols
Suppose $$H$$ is a central factor of a group $$G$$ (in other words, $$HC_G(H) = G$$). Suppose $$\varphi:G \to K$$ is a surjective homomorphism of groups. Then, $$\varphi(H)$$ is a central factor of $$K$$.

Central factor
A subgroup $$H$$ of a group $$G$$ is termed a central factor of G if it satisfies the following equivalent conditions:


 * $$HC_G(H) = G$$, where $$C_G(H)$$ denotes the centralizer of $$H$$ in $$G$$.
 * Every inner automorphism of $$G$$ restricts to an inner automorphism of $$H$$.

Similar facts about related properties

 * Transitive normality satisfies image condition
 * SCAB satisfies image condition
 * Direct factor satisfies image condition

Proof in terms of centralizers
Given: A central factor $$H$$ of a group $$G$$. A surjective homomorphism $$\varphi:G \to K$$.

To prove: $$\varphi(H)C_K(\varphi(H)) = K$$.

Proof: By the definition of homomorphism, if two elements commute in $$G$$, their images commute in $$K$$. Thus, the definition of centralizer yields:

$$\varphi(C_G(H)) \le C_K(\varphi(H))$$.

Taking the product of both sides with $$\varphi(H)$$ yields:

$$\varphi(H)\varphi(C_G(H)) \subseteq \varphi(H)C_K(\varphi(H))$$.

By the definition of homomorphism, the left side is the same as $$\varphi(HC_G(H))$$, which is $$\varphi(G)$$ (since $$H$$ is a central factor of $$G$$). $$\varphi(G) = K$$ by the assumption of surjectivity, so we get:

$$K \subseteq \varphi(H)C_K(\varphi(H)) \subseteq K$$.

This forces:

$$\varphi(H)C_K(\varphi(H)) = K$$

completing the proof.