Image-potentially direct factor equals central factor

This article gives a proof/explanation of the equivalence of multiple definitions for the term central factor
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a subgroup $H$ of a group $G$:

1. $H$ is a central factor of $G$.
2. There exists a group $L$ with a direct factor $K$ and a surjective homomorphism $f:L\to G$ such that $f(K)=H$.

Related facts

• Direct factor implies central factor
• Central factor not implies direct factor, central factor not implies complemented, complemented central factor not implies direct factor
• Central factor satisfies image condition
• Direct factor does not satisfy image condition
• Direct factor implies right-quotient-transitively central factor
• Direct factor implies join-transitively central factor
• Join of finitely many direct factors implies central factor