Left residual of 2-subnormal by normal is normal of characteristic

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

Statement

Suppose $H$ is a subgroup of a group $G$. The following are equivalent:

1. $H$ is a normal subgroup of characteristic subgroup of $G$: in other words, there is a characteristic subgroup $K$ of $G$ such that $H$ is a normal subgroup of $K$.
2. Whenever $G$ is a normal subgroup of some group $L$, $H$ is a 2-subnormal subgroup of $L$.

Related facts

• Left transiter of normal is characteristic
• Equivalence of definitions of left-transitively 2-subnormal