# Central factor satisfies image condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., image condition)
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## Statement

### 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 $K$.

## Definitions used

### Central factor

Further information: Central factor

A subgroup $H$ of a group $G$ is termed a central factor of $G if it satisfies the following equivalent conditions: * [itex]HC_G(H) = G$, where $C_G(H)$ denotes the centralizer of $H$ in $G$.

• Every inner automorphism of $G$ restricts to an inner automorphism of $H$.

## Proof

### 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.