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 $G</mah>ifitsatisfiesthefollowingequivalentconditions:*<math>HC_{G}(H)=G$, where $C_{G}(H)$ denotes the centralizer of $H$ in $G$.

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

Related facts

Similar facts about related properties

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

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:

$\varphi (C_{G}(H))\leq C_{K}(\varphi (H))$.

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.