Normal equals strongly image-potentially characteristic

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
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 normal subgroup of $G$.
2. $H$ is a strongly image-potentially characteristic subgroup of $G$ in the following sense: there exists a group $K$ and a surjective homomorphism $\rho:K \to G$ such that both the kernel of $\rho$ and $\rho^{-1}(H)$ are characteristic subgroups of $G$.

Related facts

• NPC theorem
• NRPC theorem
• Finite NPC theorem
• Finite NIPC theorem
• Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic

Facts used

1. Characteristicity is centralizer-closed

Proof

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, slight modification of NRPC theorem.