# Closure-characteristic subgroup

From Groupprops

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]

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 of a group is termed **closure-characteristic** if the normal closure of in is characteristic in .

## Relation with other properties

### Stronger properties

### 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 , both the trivial subgroup and the whole group are closure-characteristic. |

strongly join-closed subgroup property | Yes | closure-characteristicity is strongly join-closed | Suppose is a group and are all closure-characteristic subgroups of . Then the join is also closure-characteristic. |