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Core-characteristic subgroup

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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.
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Definition

QUICK PHRASES: intersection of all conjugates is characteristic, normal core is characteristic

Symbol-free definition

A subgroup of a group is termed core-characteristic if its normal core is a characteristic subgroup of the whole group.

Definition with symbols

A subgroup H of a group G is termed core-characteristic if the normal core HG of H in G is a characteristic subgroup of G.

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 click here
Automorph-conjugate subgroup every automorphic subgroup is conjugate to it click here
Intersection of automorph-conjugate subgroups intersection of automorph-conjugate subgroups
Core-free subgroup normal core is trivial
Sylow subgroup p-subgroup of finite group whose index is relatively prime to p click here
Hall subgroup subgroup of finite group whose order and index are relatively prime

Conjunction with other properties

Any normal subgroup that is also core-characteristic, is characteristic.

Incomparable properties

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