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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
QUICK PHRASES: intersection of all conjugates is characteristic, normal core is characteristic
Symbolfree definition
A subgroup of a group is termed corecharacteristic if its normal core is a characteristic subgroup of the whole group.
Definition with symbols
A subgroup of a group is termed corecharacteristic if the normal core of in is a characteristic subgroup of .
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

characteristic subgroup 
invariant under all automorphisms 


Automorphconjugate subgroup, Intersection of automorphconjugate subgroups, Intersectiontransitively automorphconjugate subgroup, Jointransitively automorphconjugate subgroupFULL LIST, MORE INFO

automorphdominating subgroup 
every automorphic subgroup is contained in a conjugate subgroup 


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automorphconjugate subgroup 
every automorphic subgroup is conjugate to it 
(via automorphdominating) 

Automorphdominating subgroup, Intersection of automorphconjugate subgroupsFULL LIST, MORE INFO

intersection of automorphconjugate subgroups 
intersection of automorphconjugate subgroups 


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corefree subgroup 
normal core is trivial 


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Sylow subgroup 
subgroup of finite group whose index is relatively prime to 


Automorphconjugate subgroup, Hall subgroup, Intersection of Sylow subgroups, Subgroup whose normal core is fully invariantFULL LIST, MORE INFO

Hall subgroup 
subgroup of finite group whose order and index are relatively prime 


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Conjunction with other properties
Any normal subgroup that is also corecharacteristic, is characteristic.
Incomparable properties