Complemented normal implies endomorphism kernel

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., complemented normal subgroup) must also satisfy the second subgroup property (i.e., endomorphism kernel)
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Statement

If $H$ is a complemented normal subgroup of $G$, $H$ is an endomorphism kernel in $G$, i.e., there is a subgroup $K$ of $G$ such that $G/H\cong K$.

Related facts

• Normal not implies endomorphism kernel
• Endomorphism kernel not implies complemented normal

Facts used

1. Complement to normal subgroup is isomorphic to quotient group

Proof

The proof is direct: the subgroup isomorphic to the quotient group is simply the permutable complement to the normal subgroup. This follows from Fact (1).

The endomorphism in question is the retraction onto that complement, i.e., the map that sends every element of the group to the unique element of the complement in its coset with respect to the normal subgroup.