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, 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

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 H_G 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 Automorph-conjugate subgroup, Intersection of automorph-conjugate subgroups, Intersection-transitively automorph-conjugate subgroup, Join-transitively automorph-conjugate subgroup|FULL LIST, MORE INFO
automorph-dominating subgroup every automorphic subgroup is contained in a conjugate subgroup |FULL LIST, MORE INFO
automorph-conjugate subgroup every automorphic subgroup is conjugate to it (via automorph-dominating) Automorph-dominating subgroup, Intersection of automorph-conjugate subgroups|FULL LIST, MORE INFO
intersection of automorph-conjugate subgroups intersection of automorph-conjugate subgroups |FULL LIST, MORE INFO
core-free subgroup normal core is trivial |FULL LIST, MORE INFO
Sylow subgroup p-subgroup of finite group whose index is relatively prime to p Automorph-conjugate subgroup, Hall subgroup, Intersection of Sylow subgroups, Subgroup whose normal core is fully invariant|FULL LIST, MORE INFO
Hall subgroup subgroup of finite group whose order and index are relatively prime |FULL LIST, MORE INFO

Conjunction with other properties

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

Incomparable properties