# Closure-characteristic subgroup

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]

## Contents

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

## Definition

QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic

A subgroup of a group is termed closure-characteristic, or a subgroup whose normal closure is characteristic, if its normal closure in the whole group is a characteristic subgroup. In symbols, a subgroup $H$ of a group $G$ is termed closure-characteristic if the normal closure $H^G$ of $H$ in $G$ is characteristic in $G$.

## 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-transitively automorph-conjugate subgroup, Join-transitively automorph-conjugate subgroup|FULL LIST, MORE INFO
automorph-dominating subgroup all automorphic subgroups are contained in conjugate subgroups |FULL LIST, MORE INFO
automorph-conjugate subgroup all automorphic subgroups are conjugate (via automorph-dominating) Automorph-dominating subgroup, Join of automorph-conjugate subgroups|FULL LIST, MORE INFO
join of automorph-conjugate subgroups join of automorph-conjugate subgroups |FULL LIST, MORE INFO
Sylow subgroup $p$-subgroup of finite group with index relatively prime to $p$ Automorph-conjugate subgroup, Hall subgroup, Join of Sylow subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
join of Sylow subgroups join of Sylow subgroups Join of automorph-conjugate subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
Hall subgroup subgroup of finite group whose order and index are relatively prime Join of Sylow subgroups, Join of automorph-conjugate subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
contranormal subgroup normal closure is whole group Subgroup whose normal closure is fully invariant|FULL LIST, MORE INFO

### Conjunction with other properties

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

## Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Metaproperty name Satisfied? Proof Difficulty level (0-5) Statement with symbols
trim subgroup property Yes (obvious) 0 In any group $G$, both the trivial subgroup $\{ e \}$ and the whole group $G$ are closure-characteristic.
strongly join-closed subgroup property Yes closure-characteristicity is strongly join-closed Suppose $G$ is a group and $H_i, i \in I$ are all closure-characteristic subgroups of $G$. Then the join $\langle H_i \rangle_{i \in I}$ is also closure-characteristic.