Direct factor of characteristic subgroup

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]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Direct factor of characteristic subgroup, all facts related to Direct factor of characteristic subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 DFC-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

Related properties

• Characteristic subgroup of direct factor