Fully invariant direct factor

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: fully invariant subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

Definition

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

1. It is both a fully invariant subgroup and a direct factor.
2. It is both a homomorph-containing subgroup and a direct factor.
3. It is both an isomorph-containing subgroup and a direct factor.
4. It is both a quotient-homomorph-containing subgroup and a direct factor.
5. It is both a normal subgroup having no nontrivial homomorphism to its quotient group and a direct factor.

Equivalence of definitions

Further information: Equivalence of definitions of fully invariant direct factor

Relation with other properties

Stronger properties

• Sylow direct factor
• Hall direct factor

Weaker properties

• Normal subgroup having no nontrivial homomorphism to its quotient group
• Quotient-homomorph-containing subgroup
• Left-transitively homomorph-containing subgroup: For full proof, refer: Fully invariant direct factor implies left-transitively homomorph-containing
• Homomorph-containing subgroup
• Fully invariant subgroup
• Isomorph-containing subgroup
• Characteristic direct factor
• Characteristic subgroup
• Direct factor