# Core-characteristic subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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
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