# Polycharacteristic 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 **polycharacteristic** in if the following holds: for any automorphism of , is a contranormal subgroup in the closure of in under the action of the cyclic subgroup generated by .

## Relation with other properties

### Stronger properties

- Characteristic subgroup
- Procharacteristic subgroup
- Weakly procharacteristic subgroup
- Paracharacteristic subgroup
- Intermediately isomorph-conjugate subgroup
- Intermediately automorph-conjugate subgroup

### Weaker properties

## Facts

- Polycharacteristic of normal implies polynormal
- Left residual of polynormal by normal equals polycharacteristic

## 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