# Paracharacteristic 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

A subgroup of a group is termed **paracharacteristic** in if for any automorphism of , is a contranormal subgroup of the subgroup .

## Relation with other properties

### Stronger properties

- Intermediately isomorph-conjugate subgroup
- Procharacteristic subgroup
- Characteristic subgroup
- Abnormal subgroup
- Weakly abnormal subgroup

### Weaker properties

- Paranormal subgroup
- Polycharacteristic subgroup
- Polynormal subgroup
- Normal-to-characteristic subgroup

## Facts

- Paracharacteristic of normal implies paranormal
- Left residual of paranormal by normal equals paracharacteristic

## Metaproperties

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties