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