Normal-isomorph-free subgroup: Difference between revisions
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Stronger than:: | | [[Stronger than::normal-isomorph-containing subgroup]] || contains any isomorphic normal subgroup of whole group || || || {{intermediate notions short|normal-isomorph-containing subgroup|normal-isomorph-free subgroup}} | ||
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| [[Stronger than:: | | [[Stronger than::characteristic-isomorph-free subgroup]] || characteristic and no isomorphic characteristic subgroup || || || {{intermediate notions short|characteristic-isomorph-free subgroup|normal-isomorph-free subgroup}} | ||
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| [[Stronger than:: | | [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || || || {{intermediate notions short|characteristic subgroup|normal-isomorph-free subgroup}} | ||
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| [[Stronger than:: | | [[Stronger than::series-isomorph-free subgroup]] || no other normal subgroup that is isomorphic and has isomorphic quotient || || || {{intermediate notions short|series-isomorph-free subgroup|normal-isomorph-free subgroup}} | ||
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| [[Stronger than:: | | [[Stronger than::normal-isomorph-automorphic subgroup]] || normal and any isomorphic normal subgroup is [[automorphic subgroups|automorphic]] || || || {{intermediate notions short|normal-isomorph-automorphic subgroup|normal-isomorph-free subgroup}} | ||
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Revision as of 14:46, 1 June 2020
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed normal-isomorph-free if it is a normal subgroup, and there is no other normal subgroup of the whole group isomorphic to it.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Isomorph-free subgroup | no other isomorphic subgroup | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal-isomorph-containing subgroup | contains any isomorphic normal subgroup of whole group | |FULL LIST, MORE INFO | ||
| characteristic-isomorph-free subgroup | characteristic and no isomorphic characteristic subgroup | |FULL LIST, MORE INFO | ||
| characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||
| series-isomorph-free subgroup | no other normal subgroup that is isomorphic and has isomorphic quotient | |FULL LIST, MORE INFO | ||
| normal-isomorph-automorphic subgroup | normal and any isomorphic normal subgroup is automorphic | |FULL LIST, MORE INFO |