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| {{conjecture|group theory}}
| | #redirect [[NIPC theorem]] |
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| ==Statement==
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| ===Verbal statement===
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| There are three versions of this conjecture, of varying degrees of strength:
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| * (''Weakest version''): Every [[normal subgroup]] of a [[group]] is an [[image-potentially characteristic subgroup]]: it arises as the image of a [[characteristic subgroup]] under a surjective homomorphism of groups.
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| * (''Intermediate version''): Every [[normal subgroup]] of a [[group]] is a [[semi-strongly image-potentially characteristic subgroup]]: there is a surjective homomorphism of groups to the whole group such that the inverse image of the subgroup is a characteristic subgroup.
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| * (''Strongest version''): Every [[normal subgroup]] of a [[group]] is a [[strongly image-potentially characteristic subgroup]]: there is a surjective homomorphism of groups to the whole group such that both the kernel of the homomorphism and the inverse image of the subgroup are [[characteristic subgroup]]s.
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| ===Statement with symbols===
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| Suppose <math>G</math> is a [[group]] and <math>H</math> is a [[normal subgroup]] of <math>G</math>. The three versions of the conjecture are:
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| * (''Weakest version''): There is a group <math>K</math>, a surjective homomorphism <math>\rho:K \to G</math>, and a ]]characteristic subgroup]] <math>L</math> of <math>G</math> such that <math>\rho(L) = H</math>.
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| * (''Intermediate version''): There exists a group <math>K</math> with a surjective homomorphism <math>\rho: K \to G</math> such that <math>\rho^{-1}(H)</math> is a [[characteristic subgroup]] of <math>K</math>.
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| * (''Strongest version''): There exists a group <math>K</math> with a surjective homomorphism <math>\rho: K \to G</math> such that both the kernel of <math>\rho</math> and the subgroup <math>\rho^{-1}(H)</math> are [[characteristic subgroup]]s of <math>K</math>.
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| The conjecture is closely related to the [[NPC conjecture]], which states that every normal subgroup is a characteristic subgroup in some bigger group.
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| ==Progress towards the conjecture==
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| ===Finite analogue is true===
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| * [[Finite NIPC theorem]]: If <math>H</math> is normal in a finite group <math>G</math>, there exists a finite group <math>K</math> and a surjective homomorphism <math>\rho: K \to G</math> such that both the kernel of <math>\rho</math> and <math>\rho^{-1}(H)</math> are characteristic subgroups of <math>K</math>.
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| * [[Finite NPC theorem]]: An analogous statement for finite groups for the closely related [[NPC conjecture]]. If <math>H</math> is normal in a finite group <math>G</math>, there exists a finite group <math>K</math> containing <math>G</math> such that <math>H</math> is characteristic in <math>K</math>.
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| ===Other results===
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| * [[No nontrivial abelian normal p-subgroup for some prime p implies every normal subgroup is strongly image-potentially characteristic]]: If <math>G</math> is a group and <math>p</math> is a prime number such that <math>G</math> does not contain any abelian normal <math>p</math>-subgroup, then every normal subgroup of <math>G</math> is a [[strongly image-potentially characteristic subgroup]] of <math>G</math>.
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| ===Results using the amalgam method===
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| A subgroup <math>H</math> of a group <math>G</math> is an [[amalgam-characteristic subgroup]] if <math>H</math> is characteristic in the [[amalgamated free product]] <math>G *_H G</math>. It turns out that [[amalgam-characteristic implies image-potentially characteristic]]. (Amalgam-characteristic subgroups need not be strongly or semi-strongly image-potentially characteristic). Thus, we get:
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| * [[Finite normal implies image-potentially characteristic]], because [[finite normal implies amalgam-characteristic]]
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| * [[Periodic normal implies image-potentially characteristic]], because [[periodic normal implies amalgam-characteristic]]
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| * [[Central implies image-potentially characteristic]], because [[central implies amalgam-characteristic]]
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| * [[Normal subgroup contained in hypercenter is image-potentially characteristic]], because [[normal subgroup contained in hypercenter is amalgam-characteristic]]
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