Fully invariant direct factor

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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:


Subgroups dissatisfying the property

Template:Subgroups dissatisfying property conjunction sorted by importance rank

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 Characteristic direct factor of nilpotent group|FULL LIST, MORE INFO
characteristic direct factor of nilpotent group the whole group is a nilpotent group |

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. |
quotient-subisomorph-containing subgroup Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO
left-transitively homomorph-containing subgroup fully invariant direct factor implies left-transitively homomorph-containing |
homomorph-containing subgroup contains every homomorphic image of itself in the whole group follows from equivalence of definitions of fully invariant direct factor Complemented homomorph-containing subgroup, Left-transitively homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO
fully invariant subgroup invariant under any endomorphism of the whole group (by definition) fully invariant not implies direct factor Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Homomorph-containing subgroup, Left-transitively homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group|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 Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Homomorph-containing subgroup|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 |
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) Characteristic direct factor, Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Homomorph-containing subgroup, Isomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Normality-preserving endomorphism-balanced subgroup|FULL LIST, MORE INFO
direct factor normal subgroup with a normal complement by definition follows from direct factor not implies characteristic Characteristic direct factor|FULL LIST, MORE INFO