Potentially normal-subhomomorph-containing equals normal

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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 equal for a subgroup H of a group G:

  1. H is a normal subgroup of G.
  2. There exists a group K containing G such that H is a Normal-subhomomorph-containing subgroup (?) of K.
  3. There exists a group K containing G such that H is a Normal-homomorph-containing subgroup (?) of K.
  4. There exists a group K containing G such that H is a Strictly characteristic subgroup (?) of K.

Proof

The same construction as used for the NPC theorem works.