Core-characteristic subgroup

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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 $H_G$ 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

• Closure-characteristic subgroup