# Characteristically DP-decomposable 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

### Definition with symbols

A subgroup inside a group is said to be characteristically DP-decomposable if we can express as an internal direct product of subgroups such that is the internal direct product of all the , and further, each is characteristic in .

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Direct product-closedness

This subgroup property is direct product-closed: it is closed under taking arbitrary direct products of groups

This actually follows from the way it's defined. The given property is the strongest direct product-closed subgroup property that is weaker than the property of being characteristic.

### 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

Clearly, the trivial subgroup, as well as the whole group, are characteristically DP-decomposable.