# Weakly procharacteristic subgroup

From Groupprops

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

### Definition with symbols

A subgroup of a group is termed **weakly procharacteristic** if for any automorphism of , the following holds: if denotes the closure of under the action of the cyclic group generated by , there exists such that .

## Relation with other properties

### Stronger properties

- Characteristic subgroup
- Intermediately isomorph-conjugate subgroup
- Intermediately automorph-conjugate subgroup
- Procharacteristic subgroup

### Weaker properties

## Effect of property operators

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)

### The intermediately operator

*Applying the intermediately operator to this property gives*: intermediately automorph-conjugate subgroup

If is a subgroup of that is weakly procharacteristic in every intermediate subgroup of containing it, then is an intermediately automorph-conjugate subgroup of . Conversely, if is intermediately automorph-conjugate in , then is weakly procharacteristic in every intermediate subgroup.

## Facts

- Any weakly procharacteristic subgroup of a normal subgroup is weakly pronormal.
`Further information: Weakly procharacteristic of normal implies weakly pronormal` - Weak procharacteristicity is the most general property for which this is true.
`Further information: Left residual of weakly pronormal by normal is weakly procharacteristic`