# Procharacteristic subgroup

From Groupprops

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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

- (
*Left-action convention*): A subgroup of a group is termed**procharacteristic**in if, for any automorphism of , there exists such that . - (
*Right-action convention*): A subgroup of a group is termed**procharacteristic**in if, for any automorphism of , there exists such that .

## Relation with other properties

### Stronger properties

### Weaker properties

- Automorph-conjugate subgroup
- Weakly procharacteristic subgroup
- Pronormal subgroup
- Weakly pronormal subgroup

## Facts

- Any procharacteristic subgroup of a normal subgroup is pronormal.
`Further information: Procharacteristic of normal implies pronormal` - A subgroup of a group is procharacteristic in if and only if whenever is normal in some group , is pronormal in .
`Further information: Left residual of pronormal by normal is procharacteristic`