Normal equals strongly image-potentially characteristic

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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

Facts used

  1. Characteristicity is centralizer-closed

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE], slight modification of NRPC theorem.