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-subisomorph-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

Examples

Extreme examples

• Every group is a fully invariant direct factor in itself.
• The trivial subgroup is a fully invariant direct factor in every group.

Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Sylow direct factor	Sylow subgroup that is also a direct factor
Hall direct factor	Hall subgroup (order and index are relatively prime to each other) that is also a direct factor
characteristic direct factor of abelian group	the whole group is an abelian group			|FULL LIST, MORE INFO
characteristic direct factor of nilpotent group	the whole group is a nilpotent group			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup having no nontrivial homomorphism to its quotient group	normal subgroup such that the only homomorphism from it to its corresponding quotient group is the trivial homomorphism.			|FULL LIST, MORE INFO
quotient-subisomorph-containing subgroup				|FULL LIST, MORE INFO
left-transitively homomorph-containing subgroup		fully invariant direct factor implies left-transitively homomorph-containing		|FULL LIST, MORE INFO
homomorph-containing subgroup	contains every homomorphic image of itself in the whole group	follows from equivalence of definitions of fully invariant direct factor		|FULL LIST, MORE INFO
fully invariant subgroup	invariant under any endomorphism of the whole group	(by definition)	fully invariant not implies direct factor	|FULL LIST, MORE INFO
isomorph-containing subgroup	contains any subgroup of the whole group that is isomorphic to it.	follows from equivalence of definitions of fully invariant direct factor		|FULL LIST, MORE INFO
characteristic direct factor	both a characteristic subgroup and a direct factor of the whole group.	follows from fully invariant implies characteristic	characteristic direct factor not implies fully invariant	|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms of the whole group.	follows from fully invariant implies characteristic	(via characteristic direct factor, or via fully invariant subgroup)	|FULL LIST, MORE INFO
direct factor	normal subgroup with a normal complement	by definition	follows from direct factor not implies characteristic	|FULL LIST, MORE INFO