# Direct factor of characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: direct factor and characteristic subgroup

View other such compositions|View all subgroup properties

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed a **direct factor of characteristic subgroup** if it satisfies the following equivalent conditions:

- It can be expressed as a direct factor of a characteristic subgroup
- It is a direct factor in its characteristic closure

### In terms of the composition operator

The subgroup property of being a direct factor of characteristic subgroup is obtained by applying the composition operator to the subgroup properties of being a direct factor and of being characteristic.

## Relation with other properties

### Stronger properties

- Characteristic subgroup
- Direct factor
- Minimal normal subgroup
- Direct factor of fully characteristic subgroup
- Direct root of characteristic subgroup

### Weaker properties

- Central factor of characteristic subgroup
- Direct factor of normal subgroup
- Normal subgroup of characteristic subgroup
- 2-subnormal subgroup
- Subnormal subgroup